Download a la Finance University Paris 1 Stochastic Calculus 2 Annie Millet

M2 - Mathématiques Appliquées
à l’Économie et à la Finance
University Paris 1
Spécialité : Modélisation et Méthodes Mathématiques
en Économie et Finance
Erasmus Mundus
Stochastic Calculus 2
Annie Millet
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Table des matières
1. Finite dimensional Itô processes . . . . . . . . . . . . . . . . . .
1
1.1. Reminders . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.2. Quadratic variation - Bracket of a local martingale . . . . . . . .
1.3. Real Itô processes. . . . . . . . . . . . . . . . . . . . . . . . .
1.4. Rd -valued Itô processes - General Itô’s formula . . . . . . . . . .
1.4.1. d-dimensional Itô processes . . . . . . . . . . . . . . . . . .
1.4.2. The general Itô formula . . . . . . . . . . . . . . . . . . . .
1.5.1. The Lévy characterizations . . . . . . . . . . . . . . . . . . .
1.5.2. The Markov property . . . . . . . . . . . . . . . . . . . . .
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1.6. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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2. Stochastic differential equations . . . . . . . . . . . . . . . . .
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2.1. Strong solution - Diffusion. . . . . . . . . . . . . . . . . . . . .
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1.5. Properties of Brownian motion. . . . . . . . . . . . . . . . . . .
2.2. Weak solution. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3. Some properties of diffusions . . . . . . . . . . . . . . . . . . .
2.3.1. Stochastic flows and the Markov property . . . . . . . . . . . .
2.3.2. Infinitesimal generator . . . . . . . . . . . . . . . . . . . .
2.3.3. Comparison Theorem . . . . . . . . . . . . . . . . . . . . .
2.4. Bessel Processes . . . . . . . . . . . . . . . . . . . . . . . . .
2.5. Diffusions and PDEs . . . . . . . . . . . . . . . . . . . . . . .
2.5.1. Parabolic problem . . . . . . . . . . . . . . . . . . . . . . .
2.5.2. The Feynman-Kac formula . . . . . . . . . . . . . . . . . . .
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2.6.1. The Sturm-Liouville - Occupation time . . . . . . . . . . . . .
2.6.2. Introduction to the Black & Sholes formula . . . . . . . . . . . .
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2.7. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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3. The Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . . .
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3.1. Changing probability . . . . . . . . . . . . . . . . . . . . . . .
3.8. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Applications to finance . . . . . . . . . . . . . . . . . . . . . . .
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4.1. Continuous financial market . . . . . . . . . . . . . . . . . . .
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2.6. Examples in finance . . . . . . . . . . . . . . . . . . . . . . . .
3.2. The Cameron Martin formula . . . . . . . . . . . . . . . . . . .
3.3. The Girsanov Theorem . . . . . . . . . . . . . . . . . . . . . .
3.4. The Novikov condition and some generalizations . . . . . . . . . .
3.5. Existence of weak solutions . . . . . . . . . . . . . . . . . . . .
3.6. Examples of applications to computations of expectation . . . . . .
3.7. The predictable representation theorem
. . . . . . . . . . . . .
4.1.1. Financial market with d risky assets and k factors . . . . . . . . .
4.1.2. Description of the strategies . . . . . . . . . . . . . . . . . .
4.1.3. Arbitrage-free condition . . . . . . . . . . . . . . . . . . . .
4.1.4. Neutral risk probability . . . . . . . . . . . . . . . . . . . .
4.2. Extended Black & Sholes model . . . . . . . . . . . . . . . . . .
4.2.1. Arbitrage-free and change of probability - Risk Premium . .
4.2.2. Complete market . . . . . . . . . . . . . . . . . . .
4.2.3. Computing the hedging portfolio in the Black & Sholes model
4.2.4. Volatility . . . . . . . . . . . . . . . . . . . . . .
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4.3. The Cox-Ingersoll-Ross model . . . . . . . . . . . . . . . . . . .
4.3.1. General Bessel processes . . . . . . . . . . . . . . . . . . . .
4.3.2. The Cox-Ingersoll-Ross model . . . . . . . . . . . . . . . . .
4.3.3. Price of a zero-coupon . . . . . . . . . . . . . . . . . . . . .
4.4. Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1
1 Finite dimensional Itô processes
The aim of this chapter is to extend the notions of Brownian motion, Itô process and
the Itô formula from dimension 1 to any finite dimension d.
1.1 Reminders
In order to make these notes as self-contained as possible, let us start by recalling some
notions already introduced in the notes of Stochastic Calculus 1. The filtration provides the
information available at any time t.
Definition 1.1 Let (Ω, F , P ) be a probability space.
(i) A filtration is an increasing family (Ft , t ≥ 0) of sub σ-algebras of F , i.e., Fs ⊂ Ft ⊂
F for s ≤ t.
(ii) We say that the filtration (Ft ) satisfies
T the usual conditions if it is :
• right continuous, i.e., Ft = Ft+ := s>t Fs .
• complete, i.e., all σ-algebras Ft contain P -null sets, which means that P (A) = 0 implies
that A ∈ F0 .
Convention. In all these notes, we are
0), P ) and we assume that the filtration
we suppose that F0 is the completion of
measurable random variables are almost
statements.
given a filtered probability space (Ω, F , (Ft , t ≥
(Ft ) satisfies the usual conditions. Furthermore,
the trivial σ-field {∅, Ω}, which implies that F0 surely constant. This will not be recalled in the
Definition 1.2 An (Rd -valued) stochastic process is a family (Xt , t ≥ 0) of random variables Xt : (Ω, F ) → (Rd , Rd ).
(i) The stochastic process (Xt ) is (Ft )-adapted if at each time t ≥ 0, Xt is measurable
from (Ω, Ft ) to (Rd , Rd ).
(ii) The stochastic process (Xt ) is progressively measurable measurable if at each time
t ≥ 0, the map (s, ω) 7→ Xs (ω) is measurable from B([0, t]) ⊗ Ft in (Rd , Rd ).
(iii) Let (Xt ) be a stochastic process. Its natural filtration is (FtX , t ≥ 0) where FtX =
σ(σ(Xs , s ∈ [0, t]), N ) where N denotes the P -null sets. If the process (Xt ) is right-continuous,
its natural filtration (FtX ) satisfies the usual conditions.
Theorem 1.3 Let (Xt ) be an Rd -valued, adapted and right-continuous stochastic process.
Then (Xt ) is progressively measurable.
The following notion of stopping time plays a crucial role in the theory.
Definition 1.4 A random variable τ : Ω → [0, +∞] is a stopping time (with respect to the
filtration (Ft )), also called (Ft )-stopping time, if {τ ≤ t} ∈ Ft fir every t ≥ 0. If τ is a
stopping time with respect to (Ft ), we set
Fτ = {A ∈ F : A ∩ {τ ≤ t} ∈ Ft , ∀t ∈ [0, +∞[}.
Finally, if (Xt ) is a (Ft )-adapted stochastic process, we set Xτ (ω) = Xτ (ω) (ω) ; if (Xt ) is
right-continuous and adapted, Xτ 1{τ <+∞} is Fτ -measurable.
The following examples of stopping times are important.
2009-2010
Stochastic Calculus 2 - Annie Millet
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Finite dimensional Itô processes
Proposition 1.5 Let (Xt , t ≥ 0) be an Rd -valued, (Ft )-adapted process and A ∈ Rd . Let us
recall that by convention, inf ∅ = +∞.
(i) If A is a closed set and (Xt ) is continuous, DA = inf{t ≥ 0 : Xt ∈ A} is a (Ft )stopping time.
(ii) If A is an open set and (Xt ) is right-continuous, then the hitting time of A, denoted
by TA = inf{t > 0 : Xt ∈ A}, is a (Ft+ )-stopping-time.
Proof. (i) Clearly, since X. is continuous and A is closed we have {DA ≤ t} = {ω :
inf s∈Q,s≤t d(Xs (ω), A) = 0} ∈ Ft , with d(x, A) = inf y∈A d(x, y).
(ii) To check that {TA ≤ t} ∈ Ft+ , it is enough to prove that {TA < t} ∈ Ft for every t.
Furthermore, if s < t and Xs (ω) ∈ A, the right-continuity of X. (ω) imply that for an open
set A, there exists ε ∈]0, t − s[ such that for all r ∈ [s, s + ε[, Xr (ω) ∈ A. Hence
{TA < t} = ∪s<t,s∈Q {Xs ∈ A} ∈ Ft .
Let us recall that TA need not be a (Ft )-stopping time.
2
Martingales are a central notion in the theory ; they enjoy a fundamental property of
constant use in finance : the process (Xt , t ∈ [0, T ]) is completely determined by its value
XT at terminal time T .
Definition 1.6 A real-valued (Ft )-adapted process X = (Xt , t ≥ 0) is a (Ft )-martingale if
• E[|Xt |] < +∞ (that is, Xt ∈ L1 (Ω)) for every t ≥ 0).
• E[Xt | Fs ] = Xs for all s ≤ t.
If X is a (Ft )-martingale such that E(Xt2 ) < +∞ for all t ≥ 0, X is called a squareintegrable martingale.
One may define martingales without referring to a filtration (Ft ). One requires that it is a
(FtX )-martingale, where (FtX ) is the natural σ-algebra of X. Clearly, any (Ft )-martingaleX
is also a (FtX )-martingale.
Finally, an Rd -valued process X = (Xti , i = 1, · · · , d, t ≥ 0) is a (Ft )-martingale if each
component (Xti , t ≥ 0), i = 1, · · · , d is a (Ft )-martingale.
Recall that a martingale (Xt ) with respect to a filtration (Ft ) which satisfies the usual
conditions has a right-continuous left-limited modification (cad-lag). Therefore, all the martingales we will consider will be right-continuous with left limits. The optional sampling
theorem extends to right-continuous martingales.
Theorem 1.7 (Optional Sampling theorem) Let M be a (right-continuous) (Ft )-martingale.
(i) Let S, T be (Ft )-stopping times bounded by a constant K, i.e., such that S ≤ T ≤ K.
Then MT is integrable and
E(MT |FS ) = MS a.s.
(ii) Let T be a stopping time. The stopped process (MtT , t ≥ 0) defined by
MtT = MT ∧t
(1.1)
is also a (Ft )-martingale.
Stochastic Calculus 2 - Annie Millet
2009-2010
1.1
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Reminders
Proof. (i) For each n ≥ 1, set Sn (ω) = k2−n on {S ∈ [(k − 1)2−n , k2−n [}, k ≤ K2n + 1 and
let Tn be defined in a similar way. Then Sn and Tn are (Ft )-stopping times taking finitely
many values and such that S ≤ Sn ≤ Tn , limn Sn = S and limn Tn = T . If A ∈ FSn .
Decomposing
according
to the values of Sn and using the martingale property,
R the set A P
R
we deduce A MSn dP = k A∩{Sn =k} MK dP . Hence MSn = E(MK |FSn ). Since FSn ⊂ FTn ,
this yields
E(MTn |FSn ) = E(E(MK |FTn )|FSn ) = MSn .
R
R
Therefore, given any set A ∈ FS ⊂ FSn one has A MSn dP = A MTn dP . Furthermore,
the sequences of stopping times (Sn , n ≥ 1) and (Tn , n ≥ 1) are decreasing and the above
computation shows that the sequences (MSn , FSn ) and (MTn , FTn ) are backward martingales,
and hence are uniformly integrable. Since the martingale M is right-continuous,
MT =
R
1
a.s.
and
in
L
.
We
deduce
that
for
any
A
∈
F
,
M
and
M
=
lim
M
lim
M
S
S dP =
S
n
Sn
A
R n Tn
MT dP , which concludes the proof. Notice that this proof easily extends when S et T
A
are non-bounded stopping times such that S ≤ T when the martingale M is uniformly
integrable.
(ii) Let s ≤ t, and apply part (i) to the stopping times s ∧ T ≤ t ∧ T ≤ t. We deduce
that Mt∧T is a (Ft∧T )-martingale. Let us check that this process is (Ft )-adapted, integrable
and is also a (Ft )-martingale.
Let A ∈ Fs . Obviously, A ∩ {T > s} ∈ Fs∧T , and since E(Mt∧T |Fs∧T ) = Ms∧T , we have
Z
Z
Mt∧T dP =
Ms∧T dP.
A∩{T >s}
A∩{T >s}
Furthermore,
on the set {T ≤ s} it holds Mt∧T = MT = Ms∧T ; we deduce
R
Ms∧T dP , which concludes the proof.
2
A
R
A
Mt∧T dP =
This proposition justifies the next definition, which makes it possible to localize the
notion of martingale by introducing an increasing sequence of stopping times.
Definition 1.8 An (Ft )-adapted right-continuous process M is an (Ft ) local martingale if
there exists an increasing sequence of (Ft )-stopping times such that τn → ∞ and for every
n, the process M τn := (Mt∧τn , t ≥ 0) is a (Ft )-martingale.
Remark 1.9 (1) Let M be a local martingale. If one replaces the sequence of stopping times
(τn ) by (τn ∧ n) we see that we may require that each of the martingales M τn is uniformly
integrable. For each n ≥ 1, let Sn = inf{t ≥ 0 : |Mt | ≥ n}. Then Sn is a stopping time and
if the local martingale M is continuous, replacing τn by τn ∧ n ∧ Sn , we may impose that the
martingale M τn remains bounded.
(2) An integrable local martingale M need not be a martingale. Such an counter-example
can be found in [11], page 182.
Definition 1.10 The process (Bt , t ≥ 0) is a (standard) real -or 1-dimensional- Brownian
motion if the following properties (a)-(c) are satisfied :
a) P (B0 = 0) = 1 (the Brownian motion starts from 0).
b) For times 0 ≤ s ≤ t, the random variable Bt − Bs is centered Gaussian with variance
(t − s), denoted by N (0, t − s).
c) For every integer n and 0 ≤ t0 ≤ t1 · · · ≤ tn , the random variables Bt0 , Bt1 − Bt0 , · · · ,
Btn − Btn−1 are independent.
2009-2010
Stochastic Calculus 2 - Annie Millet
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Finite dimensional Itô processes
Recall that the trajectories of the Brownian motion B are a.s. continuous, and even
α-Hölder functions with α < 12 , but that a.s. they are note differentiable (and do not even
1
belong the space C 2 ).
We will often suppose that the filtration (Ft ) is the natural Brownian filtration (FtB )
associated with the Brownian motion B. and hence that it satisfies the optional sampling
theorem 1.7. Property c) shows that for 0 ≤ s < t, the increment Bt − Bs is independent
from the σ-algebra FsB = σ(σ(Bu , 0 ≤ u ≤ s), N ). Therefore, the Brownian motion B is a
(FtB )-martingale.
Recall some properties of the Brownian motion which be constantly used. Their proof is
left as an exercise.
Proposition 1.11 Let (Bt ) be a Brownian motion. Then :
(i) (Scaling) For any constant c > 0, the process cBt/c2 is a Brownian motion and (−Bt )
is a Brownian motion.
(ii) (Bt2 − t, t ≥ 0) is a (FtB )-martingale.
2
(iii) For any θ ∈ R, the process exp θBt − θ2 t , t ≥ 0 is a (FtB )-martingale.
The next graphic shows three samples of Brownian trajectories of obtained by simulation.
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1.2 Quadratic variation - Bracket of a local martingale
Let us recall the following notions.
Definition 1.12 Let (Ω, F , (Ft, t ≥ 0), P ) be a filtered probability space. For a = 1, 2 and
T ∈]0, +∞] set :
Z t
a
Ha (Ft ) = h progressively measurable such that for all t ≥ 0, E
|hs | ds < ∞ ,
0
Z t
loc
a
Ha (Ft ) = h progressively measurable such that for all t ≥ 0,
|hs | ds < ∞ a.s. ,
0
Z T
T
a
Ha (Ft ) = h progressively measurable such that E
|hs | ds < +∞ .
0
Stochastic Calculus 2 - Annie Millet
2009-2010
1.2
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Quadratic variation - Bracket of a local martingale
Rt 2
In particular, let h be a cad-lag (Ft )-adapted
process.
If
for
every
t
>
0
one
has
h ds <
0 s
R
t
+∞ a.s., then h ∈ H2loc ; if E 0 h2s ds < ∞ for every t, then h ∈ H2 (Ft ), ... When the
filtration (Ft ) is the natural Brownian filtration (FtB ) of a Brownian motion B, we simply
set H1loc := H1loc (FtB ), H2loc := H2loc (FtB ), ...
The following property of stochastic integrals with respect to the Brownian motion is
fundamental.
Theorem 1.13 Let B be a Brownian motion and let (FtB )R denote its natural Brownian
t
filtration. Let h ∈ H2 ; then the process defined by t → It = 0 hs dBs is a square integrable
(FtB )-martingale with a.s. continuous trajectories.
Rt
Let h ∈ H2loc (for example a cad-lag, (FtB )-adapted process such that 0 h2s ds < +∞
Rt
a.s.) Then the stochastic integral It = 0 hs dBs can be constructed and the process (It ) is a
continuous local martingale.
Let us recall the notion of variation satisfied by indefinite deterministic integrals.
Definition 1.14 (i) Let s < t and f : [s, t] → R. The function f has bounded variation on
the interval [s, t] if V[s,t] (f ) < +∞, where
(
)
X
V[s,t] (f ) := sup
|f (ti+1 ) − f (ti )| : {s = t0 < t1 < · · · < tn ≤ t} subdivision de [s, t] .
i
The function f : [0, +∞[ is of bounded variation on [0, +∞[ if it has bounded variation on
any interval [0, T ].
(ii) The process (Xt ) is of bounded variation on [s, t] (resp. of bounded variation) if its
trajectories a.s. have bounded variation on [s, t] (resp. have a.s. bounded variation).
Rt
Obviously, if b ∈ H1loc (Ft ), then the process t → It = 0 bs ds is a.s. of bounded variaRT
tion. Indeed, for any T ≥ 0, V[0,T ] (I) ≤ 0 |b(t)| dt. Stochastic integrals behave completely
differently.
Proposition 1.15 Let (Mt ) be an a.s. continuous (Ft ) local martingale with bounded variation. Then for any t the random variable Mt is a.s. constant (and equal to M0 ).
Proof. Since M0 is a.s. constant, replacing Mt by Mt − M0 , we may and do assume that
M0 = 0.
(i) Fix T and suppose that (Mt ) is a continuous martingale with variation V[0,T ] (M)
(a.s.) bounded by C. Let ∆ = {t0 = 0 < t1 < · · · < tn = T } be a subdivision of [0, T ],
n−1
|∆| = supi=0
|ti+1 − ti | denote its mesh and for 0 ≤ k ≤ n − 1 let Xk = Mtk+1 − Mtk . Then,
E(|MT |2 ) = E(|MT − M0 |2 ) =
X
0≤k<n
E(Xk2 ) + 2
X
E(Xi Xj ).
1≤i<j≤n−1
For i < j, Xi is Fti measurable and the martingale property
implies E(Xj |Fti ) = 0 ; hence, Pn
2
E(Xi Xj ) = E(Xi E(Xj |Fti )) = 0. Hence E(|MT | ) = k=0 E(Xk2 ) ≤ CE supk |Mtk+1 − Mtk |
and since M has continuous trajectories, a.s. the function t → Mt (ω) is uniformly continuous on [0, T ]. When |∆| → 0, we deduce that supk:tk+1 ≤t |Mtk+1 − Mtk | → 0 a.s. while
2009-2010
Stochastic Calculus 2 - Annie Millet
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Finite dimensional Itô processes
supk:tk+1≤t |Mtk+1 −Mtk | ≤ C. The Dominated Convergence Theorem yields E supk |Mtk+1 − Mtk | →
0 as |∆| → 0. We deduce that E(|MT |2 ) = 0 which implies that MT = 0 a.s.
(ii) Let (Mt ) be an a.s. continuous martingale. For every n, let
τn = inf{t ∈ [0, T ] : V[0,t] (M) ≥ n} ∧ T
(using the convention inf ∅ = +∞). The definition of V[0,t] shows that it is a (Ft )-adapted,
continuous process. Proposition 1.5 (i) shows that (τn ) is a sequence of (Ft ) stopping times.
Obviously, (τn ) is increasing and converges to T . The Optional Sampling Theorem 1.7 implies
that the process (Mtτn ) = (Mt∧τn , t ≥ 0) is a (Ft )-martingale. Furthermore, by construction,
V[0,T ] (M τn ) ≤ n a.s. Let t ∈ [0, T ] ; part (i) shows that Mt∧τn = 0 a.s. Since the trajectories
of M are a.s. continuous, as n goes to infinity, we deduce that Mt = 0 a.s.
(iii) Let finally M be a continuous local martingale and let (τn ) be a sequence of stopping
times increasing to T and such that the process (Mt∧τn , t ≥ 0) is a continuous martingale
for every n. Then for every n, this martingale is of bounded variation on [0, T ] and hence
(ii) shows that is a.s. equal to 0. Using again the a.s. continuity of M and letting n → +∞
concludes the proof.
2
This proposition implies that a.s. the BrownianR motion B does not have a bounded
t
variation on [0, T ]. The stochastic integral t → 0 σs dBs cannot be defined ω by ω
and is not of bounded variation, except if it is identically equal to 0.
The « right notion » for stochastic integrals, as well as for the Brownian motion, is that
of quadratic variation. For any process X defined on [0, T ], 0 ≤ t ≤ T and any subdivision
∆ = {t0 = 0 < t1 < · · · < tk = T }, let
Tt∆ (X)
=
k−1
X
i=0
|Xti+1 ∧t − Xti ∧t |2 .
Definition 1.16 Let X : [0, T ] × Ω → R denote a stochastic process defined on [0, T ]. We
say that X has a finite quadratic variation if for every t ∈ [0, T ],
hX, Xit = lim Tt∆ exists in probability,
|∆|→0
or in other words that for any sequence (∆n ) of subdivisions de [0, T ] whose mesh convergences to 0, the sequence (Tt∆n , n ≥ 1) converges in probability to a limit denoted by hX, Xit .
The next two results relate processes with bounded variation and with finite quadratic
variation.
Proposition 1.17 Let X be a continuous process with bounded variation on [0, T ]. Then
the quadratic variation of X on [0, T ] is equal to 0.
Proof. Let ∆ = {0 = t0 < t1 < · · · < tk = T } be a subdivision of [0, T ]. Then
k−1
X
i=0
2
|Xti+1 − Xti | ≤
k−1
X
i=0
Stochastic Calculus 2 - Annie Millet
|Xti+1 − Xti | sup |Xti+1 − Xti |.
0≤i<k
2009-2010
1.2
Quadratic variation - Bracket of a local martingale
7
Since the trajectories of X are a.s. continuous on the interval [0, T ], we have sup0≤i≤k−1 |Xti+1 −
P
Xti | → 0 a.s. when |∆| → 0. Furthermore, a.s. k−1
i=0 |Xti+1 (ω) − Xti (ω)| ≤ V[0,T ] (X. (ω)) <
+∞, which concludes the proof
Notice that the previous argument and the Dominated Convergence Theorem imply that if
V[0,t] (X) ≤ C a.s. for some constant C, Tt∆ converges to 0 in L1 .
2
The next result has been proved in the lectures of Stochastic Calculus 1.
Theorem 1.18 Let B be Brownian motion. Then (Bt ) has finite quadratic variation on
any interval [0, T ] and hB, Bit = t. More precisely, when |∆| → 0, E(|Tt∆ (B) − t|2 ) → 0,
which means that the convergence holds in LR2 .
t
Furthermore, if σ ∈ H2loc , the process ( 0 σs dBs , t ≥ 0) has finite quadratic variation
Rt 2
σ ds on any interval [0, t].
0 s
We at first extend this result to continuous local martingales.
Notation Let ∆ = {t0 = 0 < t1 < t2 < · · · } be a subdivision of [0, +∞[ such that for any
t > 0, ∆ ∩ [0, t] only contains finitely many points. Similarly, for any t > 0 (adding the point
t to the subdivision if necessary) and for any process X, set
X
(1.2)
Tt∆ (X) =
(Xti+1 ∧t − Xti ∧t )2 .
i≥0
The process X has finite quadratic variation if for any t the family of processes Tt∆ (X)
converges in probability to hX, Xit when the mesh |∆| of the subdivision on [0, t] converges
to 0.
The following result is fundamental. It connects the quadratic variation of a process and
a martingale associated with the square of the process.
Theorem 1.19 Let M be a continuous (Ft )- local martingale. Then M has finite quadratic
variation and the quadratic variation process hM, Mit is the unique increasing, adapted,
continuous process null at zero, such that
Mt2 − hM, Mit , t ≥ 0 is a (Ft ) local martingale .
Furthermore, for s < t and any sequence (∆
n ) of subdivisions with mesh |∆n | converging to
0, the sequence sups≤t Ts∆n (M) − hM, Mis converges to 0 in probability.
If moreover M is a square integrable martingale (that means if E(Mt2 ) < +∞ for every
t), then (Mt2 − hM, Mit , t ≥ 0) is a (Ft ) continuous martingale such that for any pair of
bounded stopping times S ≤ T ≤ C,
E(MT2 − MS2 |FS ) = E(|MT − MS |2 |FS ) = E(hM, MiT − hM, MiS |FS ).
Proof. Uniqueness can be deduced from Proposition 1.15. Indeed, let Mt2 = Yt + At =
Zt + Bt where A, B are continuous processes with bounded variation null at 0, Y, Z are local
martingales which are continuous since the processes M.2 , A. et B. are continuous. Then the
difference Y − Z = B − A is a continuous local martingale, with bounded variation and null
at 0. Proposition 1.15 shows that is is null.
The proof of the existence will be done in several steps. To ease the notation, we will
skip the filtration when dealing with stopping times, martingales, ... Replacing (Mt ) by
(Mt − M0 , t ≥ 0) which is also a (local) martingale we may and do assume that M0 = 0.
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Finite dimensional Itô processes
(1) Suppose that M is a martingale bounded by C. For any subdivision ∆, s < t, let i
and k denote the integers such that ti ≤ s < ti+1 and tk ≤ t < tk+1 .
Then if i = k, the martingale property implies that
E Tt∆ (M) − Ts∆ (M)Fs = E(|Mt − Mti |2 − |Ms − Mti |2 |Fs )
= E(|Mt − Ms |2 |Fs ) = E(Mt2 − Ms2 |Fs ).
If i < k, then
E |Mti+1 − Mti |2 |Fs = E |Mti+1 − Ms |2 |Fs + |Ms − Mti |2 .
P 2
Using the convention nj=n
xj = 0 if n1 > n2 , we deduce
1
k−1
X
E Tt∆ (M) − Ts∆ (M)Fs = E |Mti+1 − Ms |2 +
|Mtj+1 − Mtj |2 + |Mt − Mtk |2 Fs .
j=i+1
On the other hand, for s ≤ a < b ≤ c < d ≤ t,
E[(Mb − Ma )(Md − Mc )|Fs ] = E[(Mb − Ma )E[Md − Mc |Fc ]|Fs ] = 0.
Decompose Mt − Ms as a sum of increments using the points {s, ti+1 , · · · , tk , t} ; we obtain
k−1
X
E |Mt − Ms |2 |Fs = E |Mti+1 − Ms |2 +
|Mtj+1 − Mtj |2 + |Mt − Mtk |2 Fs ,
j=i+1
which yields
E Mt2 − Ms2 |Fs = E (Mt − Ms )2 |Fs = E Tt∆ (M) − Ts∆ (M)|Fs .
(1.3)
Therefore, the process (Mt2 − Tt∆ (M), t ≥ 0) is a square integrable martingale and E(Mt2 ) =
E(Tt∆ (M)) for any t ≥ 0.
(2) Let M be a continuous martingale bounded by C. Fix a and let (∆n ) be a sequence of
subdivisions of [0, a] with mesh converging to 0. We prove that the sequence (Ta∆n (M), n ≥ 0)
converges in L2 , which means that it is a Cauchy sequence in L2 .
˜ denote the subdivision obtained using the union
Let ∆ and ∆′ be subdivisions and let ∆
′
of the points of ∆ and ∆ . Set
′
X = T ∆ (M) − T ∆ (M).
Since (Mt2 − Tt∆ (M) , t ≥ 0) is a martingale, X is also a martingale such that X0 = 0 and
(Xt , 0 ≤ t ≤ a) remains bounded. The above computation with X instead of M shows that
h
i
i
h
˜
2
∆
∆′
∆
2
E(Xa ) = E |Ta (M) − Ta (M)| = E Ta (X) .
h
i
′
˜
˜
˜
Furthermore, Ta∆ (X) ≤ 2 Ta∆ T ∆ (M) + Ta∆ T ∆ (M) .
To check that the sequence Ta∆n (M) is Cauchy, it is enough to check that
h
i
˜
∆
∆
E Ta T (M)) converges to 0 as |∆| + |∆′ | → 0.
Stochastic Calculus 2 - Annie Millet
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Quadratic variation - Bracket of a local martingale
˜ and let tl denote the unique element of ∆ such that tl ≤ sk < sk+1 ≤ tl+1 . Then,
Let sk ∈ ∆
Ts∆k+1 (M) − Ts∆k (M) = (Msk+1 − Mtl )2 − (Msk − Mtl )2 = (Msk+1 − Msk )(Msk+1 + Msk − 2Mtl ).
We deduce that
˜
Ta∆
∆
T (M) ≤
˜
Ta∆ (M)
2
sup |Msk+1 + Msk − 2Mtl | .
k
Since M is a continuous and bounded process, the dominated convergence theorem implies
4
E sup |Msk+1 + Msk − 2Mtl | → 0 when |∆| + |∆′ | → 0.
k
˜
Using Schwarz’s inequality, it remains to check that the sequence E(|Ta∆ (M)|2 ) is bounded
by some constant, which ends up to checking that
sup E(|Ta∆ (M)|2 ) < +∞.
∆
Let ∆ be a subdivision containing the point a = tn . Then
|Ta∆ (M)|2 =
=
n−1
X
i=0
n−1
X
i=0
=
n−1
X
i=0
|Mti+1 − Mti |2
!2
|Mti+1 − Mti |4 + 2
|Mti+1 − Mti |4 + 2
n−2
X
i=0
n−2
X
i=0
|Mti+1 − Mti |2
Mti+1 − Mti
n−1
X
j=i+1
2
|Mtj+1 − Mtj |2
Ta∆ (M) − Tt∆i+1 (M) .
Equation (1.3) shows that E Ta∆ (M) − Tt∆i+1 (M)|Fti+1 = E (Ma − Mti+1 )2 |Fti+1 . Since
(Mti+1 − Mti )2 is Fti+1 -measurable, we deduce
E(|Ta∆ (M)|2 )
=
n−1
X
i=0
E(|Mti+1 − Mti |4 )
n−1
X
+2
E |Mti+1 − Mti |2 E Ta∆ (M) − Tt∆i+1 (M)|Fti+1
i=0
=
n−1
X
E(|Mti+1
i=0
n−1
X
− Mti | ) + 2
E |Mti+1 − Mti |2 (Ma − Mti+1 )2
4
i=0
∆
2
2
≤ E sup |Mtk+1 − Mtk | + 2 sup |Ma − Mtk | Ta (M) .
k
k
Since supt |Mt | ≤ C and M0 = 0, equation (1.3) used with 0 and a implies E(Ta∆ (M)) ≤ C 2
and hence
E(|Ta∆ (M)|2 ) ≤ 12C 2 E(Ta∆ (M)) ≤ 12C 4 .
(1.4)
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Finite dimensional Itô processes
Therefore, the sequence (Ta∆n (M), n ≥ 1) is Cauchy in L2 ; it converges in L2 (and hence in
probability) to a limit which is denoted by hM, Mia .
(3) Let M be a continuous martingale bounded by C. We check that the process hM, Mi
satisfies the properties claimed in the statement of the theorem. Let (∆n ) be a sequence of
subdivisions with mesh |∆n | converging to 0. For m < n, the process T ∆n (M) − T ∆m (M) is
a martingale and Doob’s inequality implies
h
∆n
∆m
2
E sup |Ts (M) − Ts (M)| ≤ 4E |Ta∆n (M) − Ta∆m (M)|2 .
0≤s≤a
∆
Let m(k) be an integer such that E(|Ts∆m (M) − Ts m(k) (M)|2 ) ≤ 2−k for m ≥ m(k). We
may and do suppose that the sequence m(k) is strictly increasing. The Borel Cantelli
lemma implies that we may extract a subsequence (T.∆m(k) (M), k ≥ 1) - from the sequence
(T.∆n (M), n ≥ 1) - which a.s. converges uniformly on the interval [0, a].
Using a diagonal procedure, we may extract a further subsequence which a.s. converges
uniformly on the interval [0, N] for any integer N. Hence the limiting process hM, Mi is a.s.
continuous. Furthermore, since the limit does not depend on the sequence of subdivisions,
we may assume that this sequence is such that ∆n ⊂ ∆n+1 and that ∪n ∆n is dense in
[0, +∞[.
Let s < t be elements of ∪n ∆n ; there exists n0 such that s, t ∈ ∆n for any n ≥ n0 . Then
obviously Ts∆n ≤ Tt∆n for n ≥ n0 , so that hM, Mis ≤ hM, Mit . By continuity, we deduce
that the process hM, Mi is increasing. Finally, letting n go to infinity in (1.3)written with
the subdivision ∆n and using the uniform integrability of the sequence (Tt∆n (M), n ≥ 1)
(which is bounded in L2 by (1.4)), we deduce that M 2 − hM, Mi is a martingale.
(4) Let M be a continuous local martingale and (Tn ) a sequence of stopping times a.s.
increasing to +∞ and such that for any n the process X(n) = M Tn defined by (1.1) is a
bounded continuous martingale. The preceding proof shows that there exists an increasing
process A(n) null at 0 such that for every n, the process (X(n)2t − A(n)t , t ≥ 0) is a
martingale. Furthermore, the stopped process
(X(n + 1)2 − A(n + 1))Tn = X(n)2 − A(n)Tn
is a martingale and A(n + 1)Tn is an increasing process null at en 0. Uniqueness proved in
part (1) shows that A(n + 1)Tn = A(n)Tn a.s. This defines without ambiguity an increasing
process hM, Mit = A(n)t for every t ≤ Tn . The uniqueness is deduced from that on any
interval [0, Tn ].
Fix t > 0, ε > 0 and δ > 0. The sequence of stopping times (Tn ) goes to +∞, so that
for large enough n, S = Tn is such that P (S ≤ t) < δ and the martingale M S remains
bounded. The first part shows that as the mesh of the subdivision |∆| goes to 0, Tt∆ (M S )
converges to hM S , M S it in probability. Since Ts∆ (M S ) = Ts∆ (M) and hM S , M S is = hM, Mis
for s ∈ [0, S], we deduce that for small |∆| one has
S
S
S
∆
∆
P sup |Ts (M) − hM, Mis | ≥ ε ≤ δ + P sup |Ts (M ) − hM , M is | ≥ ε ≤ 2δ.
s≤t
s≤t
(5) Let finally M be a square integrable martingale. Then Doob’s inequality implies,
E( sup |Ms |2 ) ≤ 2E(|Mt |2 ).
0≤s≤t
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Quadratic variation - Bracket of a local martingale
11
On the other hand, let (Tn ) be an increasing sequence of stopping times a.s. converging
to +∞ and such that the process Xn = M Tn is a continuous bounded martingale for
any n. For every t, E(hM, Mit∧Tn ) = E(|Mt∧Tn |2 ) and the sequence (Mt∧Tn , n ≥ 0) is a
(Ft∧Tn , n ≥ 0) martingale bounded in L2 which converges in L2 to Mt2 . Furthermore, the
monotone convergence theorem shows that
E(hM, Mit ) = lim E(hM, Mit∧Tn ) = lim E(|Mt∧Tn |2 ).
n
n
Finally the discrete martingale (Mt∧Tn , n ≥ 1) is closed by Mt ∈ L2 and since ∨n Ft∧Tn = Ft ,
it converges in L2 to Mt . Therefore, E(hM, Mit ) = E(Mt2 ) < +∞. Doob’s inequality shows
that for s ≤ t,
Ms2 − hM, Mis ≤ sup Mr2 + hM, Mir ∈ L1 .
r≤t
Exercise 1.4 (ii) shows that this continuous, uniformly integrable local martingale is a martingale. The optional sampling theorem 1.7 concludes the proof.
2
The bracket of two continuous local martingales M and N id defined using polarization.
Theorem 1.20 Let M and N be continuous (Ft )-local martingales. There exists a unique
adapted, continuous process, with bounded variation and null at 0, denoted by hM, Ni such
that (Mt Nt −hM, Nit , t ≥ 0) is a continuous local martingale. Furthermore, for any sequence
(∆n ) of subdivisions of [0, t] with mesh converging to 0, the sequence
X
(Mti+1 ∧s − Mti ∧s )(Nti+1 ∧s − Nti ∧s ) − hM, Nis converges to 0 in probability.
sup s≤t
ti ∈∆n
Proof. Uniqueness is a straightforward consequence of proposition 1.15. To prove the existence of this process, it suffices to check that
i
1h
hM, Ni = hM + N , M + Ni − hM − N , M − Ni .
4
has this required properties. It is the difference of increasing processes and hence it has a
bounded variation.
2
Definition 1.21 The above process hM, Ni is called the bracket of M and N and the process
hM, Mi also denoted by hMi is the increasing process associated with M.
Definition 1.22 A process X is a continuous semi-martingale if for all t it can be written
as Xt = X0 +Mt +At , where (Mt ) is a continuous (Ft )-local martingale, (At ) is a continuous
process with bounded variation, and M0 = A0 = 0.
The following result is easily deduced form the above statements.
Proposition 1.23 The quadratic variation of a continuous semi-martingale X = X0 +
M + A is finite and equal to hM, Mi. The decomposition of X is unique process (up to
identification with an indistinguishable process). One denotes hX, Xi = hM, Mi and this
increasing process is called the bracket of X. Similarly, if X = X0 +M +A and Y = Y0 +N +B
are continuous semi-martingales (with continuous local martingales M, N and processes A
and B with bounded variation) the bracket of X and Y is defined as
i
1h
hX, Y i = hM, Ni = hX + Y , X + Y i − hX − Y , X − Y i .
4
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Finite dimensional Itô processes
Proof. Let X = X0 + M + A be a continuous semi-martingale. Let X = X̄0 + M̄ + Ā be
another decomposition of X, where Ā is a process with bounded variation, M̄ is a continuous
local martingale, M̄0 = Ā0 = 0. Then X̄0 = X0 , the process M − M̄ = Ā − A is a continuous
locale martingale with bounded variation. Proposition 1.15 shows that is is a.s. equal to 0.
Let ∆ be a subdivision of [0, t]. Proposition 1.17 shows that A has a null quadratic
variation and in order to prove that the quadratic variations of X and M are equal, it is
enough to check that
X
(Mti+1 − Mti )(Ati+1 − Ati ) ≤ sup |Mti+1 − Mti | V ar[0,t] (A).
i
i
The trajectories of M are a. s. continuous (and hence uniformly continuous on [0, t]) and
V ar[0,t] (A) < +∞ ; this yields that a.s. the right hand-side of the previous inequality
converges to 0 as |∆| → 0.
2
1.3 Real Itô processes.
Definition 1.24 Let (Bt )be a Brownian motion, (FtB ) its natural filtration, x ∈ R, b ∈
H1loc (FtB ) and σ ∈ H2loc (FtB ). The process X defined by
Z t
Z t
Xt = x +
σs dBs +
bs ds
(1.5)
0
0
is an Itô process. It has continuous trajectories. The process b is its drift coefficient, the
process σ its diffusion coefficient and x is the initial condition. One often denotes (1.5) as
dXt = bt dt + σt dBt ,
(1.6)
X0 = x.
Rt
Theorem 1.13 shows that Mt = 0 σs dBs is a continuous (FtB ) local martingale. TheRt
Rt
refore, an Itô process Xt = x + 0 σs dBs + 0 bs ds is a continuous local semi-martingale ;
Rt
t → 0 σs dBs is called its « martingale part » (even if it is only a local martingale) and
Rt
B
t → x+ 0 bs ds is its « bounded variation part ». The martingale part
R of Xis a « true » (Ft )t
martingale if the diffusion coefficient σ is cad-lag such that E 0 σs2 ds < +∞ for every
t > 0, or more generally of σ ∈ H2 (FtB ). It is an L2 -bounded martingale if σ ∈ H2∞ (Ft2 ).
The results of the previous section immediately yield important properties of Itô processes.
Rt
Rt
Corolary 1.25 (i) The bracket of an Itô process Xt = x + 0 σs dBs + 0 bs ds is defined by
Rt
hX, Xit = 0 σs2 ds for all t ≥ 0.
Rt
Rt
(ii) More generally, the bracket of the Itô processes Xt = x + 0 σs dBs + 0 bs ds and
Rt
Rt
Rt
Yt = y + 0 σ̄s dBs + 0 b̄s ds is that of their martingale parts, that is hX, Y it = 0 σs σ̄s ds.
Rt
Rt
Rt
Rt
(iii) Let (Xt = x + 0 σs dBs + 0 bs ds = x̃ + 0 σ̃s dBs + 0 b̃s ds, t ≥ 0) be an Itô process,
where b, b̃ ∈ H1loc (FtB ), σ, σ̃ ∈ H2loc (FtB ), x, x̃ ∈ R. Then x = x̃, b = b̃ ds ⊗ dP a.e. and
σ = σ̃ ds ⊗ dP a.e., which means that the decomposition of X is unique.
(iv) Let (Xt ) be an Itô process which is a (FtB ) local martingale. Then its drift coefficient
b is ds ⊗ dP a.e. equal to 0.
Stochastic Calculus 2 - Annie Millet
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Rd -valued Itô processes - General Itô’s formula
13
Proof. (i) and (ii) are immediate consequences of Proposition 1.17, Theorem 1.18 and of the
polarization property.
Rt
Rt
(iii) The difference Dt = 0 (b̃s − bs )ds = 0 (σs − σ̃s )dBs has both a bounded variation
on [0, T ] ( because of the deterministic integral) and a continuous local martingale (because
of the stochastic integral and of Theorem 1.13). Thus Proposition 1.17 shows that the
quadratic
R t variation2 of the stochastic integral of σ − σ̃ is null a.s. on any interval [0, t],
that is 0 |σs − σ̃s | ds = 0. This implies that σ = σ̃ ds ⊗ dP a.e. on [0, t] × Ω. Therefore,
Rt
(b − b̃s )ds = 0 for all t, which concludes the proof since X0 = x = x̃.
0 s
Rt
Rt
Rt
(iv) The Itô process Xt = x + 0 σs dBs + 0 bs ds is continuous. Since t → x + 0 σs dBs is
Rt
a continuous (FtB ) local martingale, the process t → 0 bs ds is a continuous local martingale
is of bounded variation on any interval [0, t]. Proposition 1.15 implies that it is a.s. constant,
and hence equal to 0.
2
1.4 Rd -valued Itô processes - General Itô’s formula
1.4.1 d-dimensional Itô processes
We at first extend the Brownian motion from R to Rd .
Definition 1.26 Let Bt = (Bt1 , Bt2 , . . . , Btr ), t ≥ 0 be a r-dimensional process and (Ft )
a filtration. The process B is a r-dimensional (Ft ) standard Brownian motion if the onedimensional processes (B i ), 1 ≤ i ≤ r are real-valued independent (Ft )-Brownian motions.
In other words, B0 = 0 and for 0 ≤ s ≤ t
(i) Bt − Bs is a Gaussian vector N (0, (t − s)Idr ) with mean zero and covariance matrix
(t − s)Idr .
(ii) the increment Bt − Bs is independent of Fs .
When the filtration is given explicitly, B is a standard d-dimensional Brownian if it is
a Brownian motion for its natural filtration (FtB ). If B is a standard Brownian for the
filtration (Ft ), it is also a standard Brownian for its natural filtration (FtB ). A Brownian is
a process with independent increments.
We will identify a vector (x1 , · · · , xr ) ∈ Rr and the column matrix of its components in the
canonical basis. Thus, we set


Bt1


Bt =  ...  .
Btr
We generalize Itô processes in a similar way. Let M(d, r) denote the set of d × r matrices
with d rows and r columns. The process X = (Xji (t) : 1 ≤ i ≤ d, 1 ≤ j ≤ r, t ≥ 0) taking
values in M(d, k) belongs to H1loc (Ft ) (resp. H2loc (Ft ), H22 (Ft ), H2∞ (Ft )) if each component
Xki is a real-valued process which belongs to H1loc (Ft ) (resp. H2loc (Ft ), H22 (Ft ), H2∞ (Ft )).
Definition 1.27 Let B be a r-dimensional (Ft ) standard Brownian, σ = (σki : 1 ≤ i ≤
d, 1 ≤ k ≤ r) : Ω × [0, +∞[→ M(d, r) ∈ H2loc , b = (b1 , · · · , bd ) : Ω × [0, +∞[→ Rd ∈ H1loc ,
and x = (x1 , · · · , xd ) ∈ Rd . The process (Xt ) taking values in Rd is an Itô process with
initial condition x, diffusion coefficient σ and drift coefficient b if for every i = 1, · · · , d,
Z t
r Z t
X
i
k
i
i
σk (s)dBs +
bi (s)ds.
(1.7)
Xt = x +
k=1
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Finite dimensional Itô processes
Using matrix notation, identifying a vector x = (x1 , · · · , xd ) in Rd with the column matrix
if its components in the canonical basis, one can rewrite equation (1.7) as follows
Xt = x +
Z
t
σ(s)dBs +
0
Z
t
b(s)ds,
0
where


x1


x =  ...  ,
xd

Xt1


Xt =  ...  ,
Xtd



σ11 (s) · · · σr1 (s)

..
..  ,
σ(s) =  ...
.
. 
d
d
σ1 (s) · · · σr (s)
Let us consider one-dimensional Itô processes
ξt = x +
r Z
X
k=1
t
σk (s)dBsk
0
+
Z
t
ξ¯t = x̄ +
b(s)ds and
0
r
X


b1 (s)


b(s) =  ...  .
bd (s)
σ̄j (s)dBsk
+
k=1
Z
t
b̄(s)ds,
0
Rt
for b, b̄ ∈ H1loc and σk , σ̄k ∈ H2loc . Then the process 0 b(s)ds has continuous trajectories and
P Rt
is of bounded variation, while the process rk=1 0 σk (s)dBsk is a continuous local martingale as sum of continuous local martingales. The processes ξ and ξ¯ are semi-martingales.
In
their brackets, note
order to compute
that for every index k = 1, · · · , r, the process
2 R
R
t
t
0 σk (s)dBsk − 0 |σk (s)|2 ds , t ≥ 0 is a (Ft ) local martingale.
Let k 6= l ; suppose at first that the processes σk and σl are step processes, that is
σk =
n−1
X
ξki 1]ti ,ti+1 ]
and σl =
i=0
n−1
X
i=0
ξli 1]ti ,ti+1 ] , t0 = 0 < t1 < · · · and ξki , ξlj Fti -measurables.
Let s < t ; without loss of generality, we may and do assume that the instants s and t belong
to the list of the ti , with s = tI and t = tn . Then,
E
Z
t
0
σk (u)dBuk
Z
0
t
σl (u)dBul Fs
=
=
n−1 X
n−1
X
E
i=0 j=0
Z s
0
ξki ξlj
σk (u)dBuk
Z
0
[Btki+1
−
Btki ][Btlj+1
s
σl (u)dBul .
−
Btlj ]Fs
Indeed, the independence of Fti , Btki+1 − Btki and Btli+1 − Btli , implies for example :
if I ≤ i = j, since E[(Btki+1 − Btki )(Btli+1 − Btli )|Fti ) = E[(Btki+1 − Btki )(Btli+1 − Btli )] = 0,
one deduces E(ξki ξli E[(Btki+1 − Btki )(Btli+1 − Btli )|Fti )FtI ) = 0.
The independence of B l (tj+1 ) − B l (tj ) and Ftj yields :
if i < I ≤ j, then one has ξki (Btki+1 − Btki )E(ξlj E(Btlj+1 − Btlj |Ftj )|FtI ) = 0,
if I ≤ i < j, then one has E(ξki (Btki+1 − Btki )ξlj E(Btlj+1 − Btlj |Ftj )FtI ) = 0,
This property can be extended to processes σk and σl of H22 (Ft ) for which one deduces that
r Z t
r Z t
2 X
X
2
k
|σk (s)| ds , t ≥ 0
σk (s)dBs −
k=1
0
Stochastic Calculus 2 - Annie Millet
k=1
0
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Rd -valued Itô processes - General Itô’s formula
15
is a (Ft )-martingale. A localization argument shows that this process is a (Ft ) local martingale when one only knows that σk , 1 ≤ k ≤ r belong to H2loc (Ft ).
The bracket
the processes ξ and P
ξ¯ is that
R t of their kmartingale parts
Pr Rof
t
r
k
mt = k=1 0 σk (s)dBs , and m̄t = k=1 0 σ̄k (s)dBs , that is
¯ t = hm, m̄it =
hξ , ξi
Z tX
r
σk (s)σ̄k (s)ds.
0 k=1
The bracket of the local martingale m is equal to that of ξ, that is
hξ , ξit = hm, mit =
Z tX
r
0 k=1
2
|σk (s)| ds =
Z
0
t
kσ(s)k2 ds,
where kσ(s)k denotes the Euclidian norm in Rr of the vector (σ1 (s), · · · , σr (s)).
1.4.2 The general Itô formula
The results of the previous section are gathered in the following
Proposition 1.28 Let (Bt ) be a r-dimensional standard (Ft )-Brownian motion, (Xt ) a
d-dimensional Itô processRof the form (1.7). Then X has continuous trajectories. For any
t
i = 1, · · · , d, the process ( 0 bi (s)ds, t ≥ 0) is continuous with bounded variation (that means
Rt
that each of its component is of bounded variation), null at time 0. The process 0 σ(s)dBs
is a continuous local martingale (that means that each component is a continuous local
martingale) null at time 0. The decomposition is unique. The bracket of the components X i
and X j , 1 ≤ i, j ≤ d is
Z tX
r
i
j
hX , X it =
σki (s)σkj (s)ds.
0 k=1
The Itô formula, proved in the lectures of Stochastic Calculus 1, can be generalized to
multidimensional Itô processes. Let us at first recall its simplest version in dimension one.
Rt
Rt
Theorem 1.29 Let Xt = x + 0 σ(s)dBs + 0 b(s)ds, where x ∈ R, (Bt ) is a (Ft )-Brownian
motion, b ∈ H1loc , σ ∈ H2loc . Let f : R → R be a function of class C 2 . Then for any t ≥ 0,
Z
t
Z
t
1 ′′
f (Xs )dhX, Xis
(1.8)
0
0 2
Z t
Z t
1 ′′
2
′
′
= f (x) +
f (Xs )σ(s)dBs +
f (Xs )b(Xs ) + f (Xs )σ (s) ds.
2
0
0
f (Xt ) = f (x) +
′
f (Xs )dXs +
Formally, one can set
1
df (Xt ) = f (Xt )dXt + f ′′ (Xt )dhX, Xit
2
′
where hX, Xit =
Z
t
σ 2 (s)ds.
0
The Itô formula has the following formulation in arbitrary dimension. Its proof, which
similar to that in dimension one, is omitted. Denote A∗ the transposed of the matrix A.
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1
Finite dimensional Itô processes
Theorem 1.30 Let B be a standard r-dimensional (Ft ) Brownian, σ a process taking values
in M(d, r) which belongsR to H2loc (Ft ), Rb an Rd -valued process which belongs to H1loc (Ft ) and
t
t
x ∈ Rd . Let Xt = x + 0 σ(s)dBs + 0 b(s)ds denote the Itô process defined by equations
(1.7) for i = 1, · · · , d and let f : Rd P
→ R be of class C 2 . Then if a(s) = σ(s)σ(s)∗ denotes
the d × d matrix defined by ai,j (s) = rk=1 σki (s)σkj (s) for i, j ∈ {1, · · · , d}, on a :
Z
Z tX
d
d
∂f
1 t X ∂2f
i
(Xs )dXs +
(Xs )dhX i, X j is
f (Xt ) = f (x) +
∂x
2
∂x
∂x
i
i
j
0 i,j=1
0 i=1
Z tX
Z tX
k X
d
d
∂f
∂f
i
j
= f (x) +
(Xs )σj (s)dBs +
(Xs )bi (s)ds
0 j=1 i=1 ∂xi
0 i=1 ∂xi
Z
d
1 t X ∂2f
+
(Xs )ai,j (s)ds.
2 0 i,j=1 ∂xi ∂xj
Let
∂2f
∂x2
d
denote the symmetric matrix
∂2f
,1
∂xi ∂xj
(1.9)
(1.10)
≤ i, j ≤ d , (., .) denote the scalar pro-
denote the column matrix whose components are the partial derivatives
duct in R and ∂f
∂x
∂f
Then the Itô formula has the formal expression
∂xi
df (Xt ) =
∂f
1
∂2f
∗
(Xt ), dXt + T race σ(t)σ (t) 2 (Xt ) dt.
∂x
2
∂x
If the function f also depends on time, we have the second formulation of the Itô formula.
Theorem 1.31 Let B be a standard r-dimensional (Ft ) Brownian, σ a process taking values
in M(d, r) which belongsR to H2loc (Ft ), Rb an Rd -valued process which belongs to H1loc (Ft ) and
t
t
x ∈ Rd . Let Xt = x + 0 σ(s)dBs + 0 b(s)ds denote the Itô process defined by equations
(1.7) for i = 1, · · · , d and let f : [0, +∞[×Rd → R be of class C 1,2 , that means of class C 1
with respect to the time variable t and of class C 2 with respect to the space variable x. Then
if one sets a(s) = σ(s)σ(s)∗ , one has :
f (t, Xt ) = f (0, x) +
Z
t
Z
t
0
∂f
(s, Xs )ds +
∂t
d
X
Z tX
d
∂f
(s, Xs )dXsi
∂x
i
0 i=1
∂2f
(s, Xs )dhX i , X j is ds
(1.11)
∂x
∂x
i
j
0 i,j=1
Z tX
r X
d
∂f
= f (0, x) +
(s, Xs )σki (s)dBsk
∂x
i
0 k=1 i=1
#
Z t"
d
d
2
X
X
∂
f
∂f
∂f
1
+
(s, Xs ) +
(s, Xs )bi (s) +
(s, Xs )ai,j (s) ds.
∂t
∂x
2
∂x
∂x
i
i
j
0
i,j=1
i=1
+
1
2
Again, equation (1.11) can be written in a formal and compact way as follows :
∂f
1
∂2f
∂f
∗
(t, Xt )dt +
(t, Xt ), dXt + T race σ(t)σ (t) 2 (t, Xt ) dt.
df (t, Xt ) =
∂t
∂x
2
∂x
Stochastic Calculus 2 - Annie Millet
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Properties of Brownian motion.
A useful particular case of the previous formula is an « integration by parts formula » .
We leave the poof as an exercise.
Let f : [0, +∞[→ R be of class C 1 and (Bt ) be a real-valued standard Brownian motion ;
then
Z
Z
t
0
t
f (s)dBs = f (t)Bt −
Bs f ′ (s)ds.
0
The Itô formula is a very useful tool. One of its consequences is the following inequality
giving upper estimates of the moments of stochastic integrals in terms of moments of the
integrand. Let B be a real-valued standard Brownian motion,
X be a continuous (FtB )R
T
adapted process, p ∈ [1 + ∞[ and T > 0 be such that E 0 |Xs |2p ds < +∞.Then
E
Z
0
T
2p !
Z
p
p−1
≤ [p(2p − 1)] T
E
Xs dBs 0
T
2p
|Xs | ds
(1.12)
This very powerful inequality is proved in exercise 1.8. The following result strengthens the
conclusion. The right inequality is proved in exercise 1.8.
Theorem 1.32 (The Burkholder-Davies-Gundy theorem) For any p ∈ [1, +∞[ there exist
universal constants kp > 0 and Kp > 0 (which only depend on p) such that for any continuous
square integrable (Ft )-martingale M and any T > 0,
p
2p
≤ Kp E (hMipT ) .
kp E (hMiT ) ≤ E sup |Ms |
(1.13)
0≤s≤T
1.5 Properties of Brownian motion.
1.5.1 The Lévy characterizations
Note that given a d-dimensional standard Brownian motion B, for every i, j = 1, · · · , d
the processes (Bti , t ≥ 0) and (Bti Btj − δi,j t , t ≥ 0) are (FtB )-martingales (with δi,j = 0
for i 6= j and δi,i = 1). This is proved in exercise 1.7 We next prove that these martingale
properties characterize the Brownian motion, which will play a crucial role to study changes
of probability. Let (u, v) denote the scalar product of the vectors u, v ∈ Rd and kuk denote
the Euclidean norm of u,
Theorem 1.33 (Paul Lévy’s characterization) Let X = (Xt = (Xt1 , · · · , Xtd ), t ≥ 0) be an
(Ft )-adapted processes taking values in Rd .
(i) Suppose that for any u ∈ Rd ,
1
2
E ei(u,Xt −Xs ) | Fs = e− 2 kuk (t−s) .
(1.14)
Then (Xt ) is a standard (Ft )-Brownian motion taking values in Rd .
(ii) Suppose that for every u ∈ Rd , the processes X and t 7→ exp [(u, Xt ) − kuk2t/2] are
(Ft )-martingales. Then X is a standard (Ft )-Brownian motion taking values in Rd .
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Finite dimensional Itô processes
(iii) Suppose that the process defined by Mtj = Xtj − X0j for j = 1, · · · , d is a (Ft ) local
martingale null at 0 (i.e., M0i = 0 for every i) and that the brackets of M i and M j are
hM i , M j it = δi,j t.
(1.15)
Then M is a standard (Ft )-Brownian motion taking values in Rd .
Proof. (i) One has to prove that for 0 ≤ s < t the random vector Xt − Xs is Gaussian
N (0, (t − s)Id) and independent of Fs .
Let Z be a Gaussian vector N (0, (t − s)Id). Using condition (1.14), we have
kuk2 (t−s)
E ei(u,Xt −Xs ) |Fs = E ei(u,Z) = e− 2 .
(1.16)
Indeed, fix u ∈ Rd and set Φ(x) = ei(u,x) ; for any set A ∈ Fs we have E[1A Φ(Xt −
Xs )] = P (A)E[Φ(Z)]. Since the characteristic function characterizes the distribution, we
deduce
that for any bounded Borel function f : Rd → R and any set A ∈ Fs , we have
E 1A f (Xt − Xs ) = P (A)E[f (Z)]. This proves that Xt − Xs is independent of Fs and also
that the distribution of Xt − Xs is equal to that of Z.
(ii) Using part (i), it is enough to show that (1.16) holds. Let at first v ∈ Rd ; by
assumption, for 0 ≤ s < t,
kvk2t
kvk2 t E exp (Xt − Xs , v) | Fs = exp −(Xs , v) +
E exp (Xt , v) −
Fs
2
2
kvk2 (t − s)
= exp
.
2
An induction argument (based on the fact that both hand sides of (1.16) are analytic
functions of one of the variables vk ∈ C, the other variables being fixed either in R or in
C) proves that the previous identity can be extended from v ∈ Rd to v ∈ Cd . Using this
equality with the vector v with components vk = iuk shows that (1.16) holds.
(iii) We only prove this
R t characterization in the particular case if an Itô process (Xt ),
that is when Xt = X0 + 0 H(s)dBs with H ∈ H2loc . The general case requires a notion
of stochastic integral more general than that with respect to the Brownian motion defined
in the lectures of Stochastic Calculus
1. For j = 1, · · · , d and t > 0, the
bracket of X j at
R
P
P
t
any time t is hX j , X j it = t = rk=1 0 |Hkj (s)|2 ds, which implies that rk=1 |Hkj (s)|2 = 1
for almost every s. Furthermore, if j 6= l, then
t > 0, the bracket hX j , X l it =
Ra.s.
Pr for any
t Pr
j
j
l
l
k=1 Hk (s)Hk (s) ds = 0, which implies that
k=1 Hk (s)Hk (s) = 0 a.s. for almost every
0
s. We deduce that H ∈ H22 and that (Xt , 0 ≤ t ≤ T ) is a martingale. The above inequalities
prove that the vectors (H j (s), 1 ≤ j ≤ d) are an orthonormal family (and hence a linear
d
independent family)
of Rr , which yields
d ≤ r. For any set A ∈ Fs and any vector λ ∈ R ,
let f (t) = E 1A exp [i(λ, Mt − Ms )] . Using part (i), it is enough to prove that f (t) =
2
P (A) exp − kλ2 k (t − s) . The Itô formulaused separately
for the real and the imaginary
Pd
1
d
j j
part of the function x = (x , · · · x ) → exp i j=1 λ x , implies for s < t,
i(λ,Mt )
e
i(λ,Ms )
= e
+i
d X
r
X
j=1 k=1
−
Z
d X
r
X
λj λl
j,l=1 k=1
Stochastic Calculus 2 - Annie Millet
2
s
t
λ
j
Z
s
t
ei(λ,Mu ) Hkj (u)dBuk
ei(λ,Mu ) Hkj (u)Hkl (u)du,
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Properties of Brownian motion.
which yields
i(λ,Mt −Ms )
e
=1+i
r
d X
X
j=1 k=1
λ
j
Z
t
i(λ,Mu −Ms )
e
s
Hkj (u)dBuk
−
Z
d
X
|λj |2
j=1
2
t
ei(λ,Mu −Ms ) du.
s
For any j = 1, · · · , d and k = 1, · · · , r the stochastic integrals (Nkj (t), t ≥ s) defined by
Rt
Nkj (t) = s ei(λ,Mu −Ms ) Hkj (u)dBuk for t ≥ s are martingales such that E(Njk (t)|Fs ) = 0. The
Fubini theorem implies that
Z t
Z
kλk2
kλk2 t
i(λ,Mu −Ms )
f (t) = P (A) −
E 1A
f (u)du.
e
du = P (A) −
2
2
s
s
2
Since f ′ (t) = − kλk
f (t) for t ≥ s and since f (s) = P (A), we have proved that for any t ≥ s,
2
kλ2 k
f (t) = P (A) exp(− 2 (t − s)).
2
1.5.2 The Markov property
A fundamental property of the Brownian motion is the Markov property, which says
that what happens in the future at time t does not really depend on the whole past, that is
on the whole σ-algebra Ft of the « past » , but only of the state of Bt at time t. We at first
describe how one moves from time s to time t ≥ s.
Definition 1.34 A transition probability on Rd is a map Π : Rd × Rd → [0, 1] such that
(i) for any x ∈ Rd , the map A ∈ Rd → Π(x, A) is a probability.
(ii) for any A ∈ Rd , the map x ∈ Rd → Π(x, A) is measurable from (Rd , Rd ) to
([0, 1], B([0, 1]).
A transition function on Rd is a family (Ps,t , 0 ≤ s < t) of transition probabilities such that
for s < t < v the Chapman-Kolmogorov is satisfied :
Z
Ps,t(x, dy)Pt,v (y, A) = Ps,v (x, A) , ∀A ∈ Rd , ∀x ∈ Rd .
(1.17)
For any positive (or bounded) Borel function f : Rd → R we set
Z
Ps,t f (x) =
f (y) Ps,t(x, dy).
Rd
If the transition function Ps,t only depends on the difference t−s, it is said to be homogeneous
and then, if one sets Pt = P0,t , the equation (1.17) can be written as follows
Z
Ps+t (x, A) = Ps (x, dy)Pt (y, A).
For any positive (or bounded) Borel function f : Rd → R, one has Ps+t f (x) = Pt Ps f (x)
and (Pt ) is called a semi-group.
The equation (1.17) est natural. Indeed, if X is a process and (x, A) ∈ Rd × Rd →
Ps,t (x, A) is a transition probability such that for A ∈ Rd and s < t,
P (Xt ∈ A|σ(Xu , u ≤ s)) = P (Xt ∈ A|Xs ) a.s.
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1
and
P (Xt ∈ A|Xs = x) = Ps,t (x, A) =
Z
Finite dimensional Itô processes
1A (y) Ps,t(x, dy),
we deduce that Ps,t (x, dy) is a transition probability (since
R it is the conditional distribution
of Xt given Xs = x) and that E(f (Xt )|σ(Xu , u ≤ s)) = Rd f (y)Ps,t(Xs , dy) for any positive
(or bounded) Borel function f : Rd → R. Given s < t < v, A ∈ Rd and f (y) = Pt,v (y, A),
we deduce that
Ps,v (Xs , A) = P (Xv ∈ A|σ(Xu , u ≤ s))
= E P (Xv ∈ A|σ(Xu , u ≤ t) | σ(Xu , u ≤ s)
Z
= E f (Xt )|σ(Xu , u ≤ s) = Ps,t (Xs , dy)Pt,v (y, A).
These notions give a precise formulation of the « weak » Markov property.
Definition 1.35 (1) The Rd -valued, (Ft )-adapted process (Xt , t ≥ 0) is a Markov process
for the filtration (Ft ) if for any bounded Borel function f : Rd → R,
E f (Xt ) | Fs) = E f (Xt ) | Xs) pour tout s ≤ t.
(1.18)
(2) The Rd -valued, (Ft )-adapted process (Xt , t ≥ 0) is a Markov process for the filtration
(Ft ) with transition function Ps,t if for any bounded Borel function f : Rd → R,
Z
E f (Xt ) | Fs = Ps,t f (Xs ) :=
Ps,t (Xs , dy)f (y).
(1.19)
Rd
We say that the Markov process is homogeneous if its transition probability is homogeneous.
We prove that the Brownian motion is a homogeneous Markov process with transition
semi-group Ph defined by
Z
1
ky − xk2
Ph (x, A) = P (Bs+h ∈ A|Bs = x) =
exp −
dy,
d
2h
(2πh) 2 A
for s ≥ 0, h > 0, x ∈ Rd and A ∈ Rd , that is Ph (x, dy) is the distribution of a Gaussian
vector N (x, hId).
Theorem 1.36 (Weak Markov property) Let (Bt ) be a standard d-dimensional Brownian
motion and f : Rd → R be a bounded Borel function. Then for s ≤ t,
E[f (Bt ) | FsB ] = E[f (Bt ) | Bs ]
Z
Z
1
ky − Bs k2
dy = f (y)Pt−s(Bs , dy).
=
f (y) exp −
d
2(t − s)
(2π(t − s)) 2 Rd
Proof. The proof uses the following lemma :
Lemma 1.37 Let F be a σ-algebra, let G be a sub σ-algebra of F , and let X : (Ω, G) →
(E1 , E1 ) be a G-measurable map, Y : (Ω, F ) → (E2 , E2 ) be F -measurable map independent
of G. Then for any bounded measurable function Φ : (E1 × E2 , E1 ⊗ E2 ) → (R, R) one has
E(Φ(X, Y )|G) = ϕ(X), where ϕ : (E1 , E1) → (R, R) is defined by ϕ(x) = E[Φ(x, Y )].
Stochastic Calculus 2 - Annie Millet
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Properties of Brownian motion.
Proof of the Lemma For any measurable map V : (Ω, F ) → (E, E), let P(V ) denote
the image measure of P by VR. The definition of P(Y ) and Fubini’s theorem imply that ϕ
satisfies the equation ϕ(x) = E2 Φ(x, y)dP(Y ) (y), is bounded and measurable from (E1 , E1 )
to s (R, R). Furthermore, if Z : (Ω, G) → (R, R) is bounded and G-measurable, since the
random variables Y and (X, Z) are independent, the Fubini theorem yields
Z Z
E Φ(X, Y )Z =
Φ(x, y) z dP(X,Z) (x, z) dP(Y ) (y)
Z Z
=
Φ(x, y)dP(Y ) (dy) z dP(X,Z) (x, z)
Z
= ϕ(x) zdP(X,Z) (x, z) = E ϕ(X)Z ,
which concludes the proof of the lemma.
2
Let s < t ; apply the previous lemma with E1 = E2 = Rd , G = Fs , F = Ft , X = Bs
and
function f : Rd → R, we have
Y = Bt − Bs . We deduce that for any bounded Borel
E f (Bt )|Fs = ϕ(Bs ) where ϕ(x) = E f (x + Bt − Bs ) . A similar computation made with
G = σ(B
s ) and F = Ft and for the same random variables X = Bs and Y = Bt − Bs implies
that E f (Bt )|Bs = ϕ(Bs ). The distribution of Bt − Bs is Gaussian N (0, (t − s)Id), which
concludes the proof.
2
Corolary 1.38 Let τ be as a.s. finite (Ft ) stopping time and B be a (Ft ) standard Brownian
motion of dimension d. Then the process (Mt = Bτ +t −Bτ , t ≥ 0) is a (Fτ +t , t ≥ 0) standard
Brownian motion independent of Fτ .
Proof. Suppose at first that τ ≤ C a.s. and set Gt = Fτ +t for all t ≥ 0. Then (Gt ) is a
filtration and the optional sampling theorem 1.7 shows that if (Xt ) is a (Ft )-martingale
and if Yt = Xτ +t , then the random variable Yt is integrable and that for s < t, E(Yt |Gs ) =
E(Xτ +t |Fτ +s ) = Xτ +s = Ys , that is, (Yt ) is a (Gt )-martingale.
We have to prove that for 0 ≤ s < t, the random vector Mt − Ms is Gaussian N (0, (t −
s)Id) an independent of Gs . An argument similar to that used in the proof of the Levy
characterization of the Brownian motion shows that is suffices to check that for any vector
u ∈ Rd the equation (1.16) holds.
Let u ∈ Rd and set f (x) = ei(u,x) for x ∈ Rd . Then |f | = 1 ; decompose f into its real
and imaginary part, apply the Markov property and the definition of the Brownian motion.
We obtain
(t − s)kuk2
E(exp(i(u, Bt )|Fs ) = E(exp(i(u, Bt )|Bs ) = exp(i(u, Bs )) exp −
.
2
This yields that the process Xt = exp i(u, Bt ) + tkuk2 /2) is a (Ft )-martingale, and the
above remark shows that the process (Yt = Xτ +t , t ≥ 0) is a (Gt )-martingale. Thus, we
deduce that for all s ≤ t
Yt E(Yt |Gs )
Ys
E
= ,
Gs =
Y0
Y0
Y0
which implies that YY0t = exp(i(u, Mt ) + tkuk2 /2), t ≥ 0 is a (Gt )-martingale. Therefore,
given s ≤ t,
tkuk2
(t−s)kuk2
tkuk2
i(u,Mt −Ms ) i(u,Mt )+ 2 E e
Gs e−i(u,Ms )− 2 = e− 2 ,
Gs = E e
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Finite dimensional Itô processes
which concludes the proof of (1.16) when the stopping time τ is a.s. bounded.
Let τ be an a.s. finite stopping time ; to conclude the proof, it suffices to write (1.16) for
the sequence of bounded stopping times τ ∧ n and to let n go to +∞.
2.
This yields a result similar to the weak Markov property stated in Theorem 1.36, replacing fixes times by stopping times.
Theorem 1.39 (Strong Markov property) Let B be a Rd -valued standard (Ft )-Brownian
motion and τ be an a.s. finite (FtB ) stopping time ; then for all t > 0,
Z
1
ky − Bτ k2
dy. (1.20)
E[f (Bτ +t ) | Fτ ] = E[f (Bτ +t ) | Bτ ] =
f (y) exp −
d
2t
(2πt) 2 Rd
Proof. Apply Lemma 1.37 with X = Bτ , Y = Bτ +t − Bτ , G = Fτ (or G = σ(Xτ )), F =
Ft+τ and Corollary 1.38. We deduce that since (Bτ +t − Bτ , t ≥ 0) is a Brownian motion
independent of Fτ , the strong Markov property (1.20) holds true.
2
1.6 Exercises
Exercise 1.1 Show that a stopped local martingale M is a local martingale.
Exercise 1.2 Let (Mt ) be an integrable, cad-lag (Ft )-adapted process. Show that it is a
(Ft )-martingale if and only if for any bounded stopping time T one has MT ∈ L1 and
E(MT ) = E(M0 ).
Exercise 1.3 Let (Xt , 0 ≤ t ≤ T ) be a (Ft )-supermartingale, that is for 0 ≤ s ≤ t ≤ T ,
Xs ≥ E(Xt |Fs ), and suppose that E(XT ) = E(X0 ). Show that (Xt ) is a martingale.
Exercise 1.4 Let (Mt ) be a (Ft ) local martingale. Show that
(i) If Mt is non-negative, then it is a supermartingale.
(ii) If there exists Y ∈ L1 such that for all t, |Mt | ≤ Y a.s., then (Mt ) is a martingale.
Exercise 1.5 Let B be a real Brownian motion and (FtB ) denote its natural
filtration.
Rt
Characterize the values of α ∈ R such that the process defined by It = 0 |Bs |α dBs is a
square-integrable (FtB )-martingale.
Rt
Rt
Exercise 1.6 Let B be a (Ft )-Brownian motion, Xt = x + 0 σs dBs + 0 bs ds and X̄t =
Rt
Rt
x̄ + 0 σ̄(s)dBs + 0 b̄s ds be Itô processes. Write Xt Yt as a semi-martingale.
Exercise 1.7 Let (Bt1 , · · · , Btr ) be a standard r-dimensional Brownian motion.
1. Let X and Y be real-valued random variables, G be a sub σ-algebra of F such that
the σ-algebras σ(X), σ(Y ) and G are independent. Show that E(XY |G) = E(X)E(Y )
a.s.
2. Prove that for s < t, when i 6= j one has E(B i Btj Fs ) = B i B j .
t
i
s
s
j
3. Prove that hB , B it = δi,j t.
4. Prove that for every u = (u1 , · · · , ur ) ∈ Rr ,
1
1
2 2
E e(u,Bt )− 2 kuk t Fs = e(u,Bs )− 2 kuk s .
Stochastic Calculus 2 - Annie Millet
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1.6
23
Exercises
R
T
0
2p
Exercise 1.8 Let (Xt ) be an (Ft )-adapted process and T > 0 be such that E
|Xs | ds <
+∞ for any p ∈ [1, +∞[.
Rt
1. Apply Itô’s formula to the process Mt = 0 Xs dBs to deduce that there exists an
increasing sequence (Tn ) if stopping times which goes to +∞ and such that
Z T
2p
2(p−1) 2
E(MT ∧Tn ) = p(2p − 1)E
Xt∧Tn dt .
|Mt∧Tn |
0
2. Is the function t → E(|Mt∧Tn |2p ) monotone ? Deduce that
Z T ∧Tn
p
2p
2p
p−1
|Xs | ds
E
E(|MT ∧Tn | ) ≤ [p(2p − 1)] T
0
and then that
2p
p
E(|MT | ) ≤ [p(2p − 1)] T
p−1
E
Z
T
2p
0
|Xs | ds .
3. Apply Doob’s inequality and show that there exists a constant Kp , which will be
written explicitly, such that
Z T
2p
2p
≤ Kp E
|Xs | ds .
E sup |Mt |
0≤s≤T
0
Exercise 1.9 Let (Xt ) be a continuous, nonnegative (Ft )-adapted process such that X0 = 0,
(At ) is an increasing, continuous (Ft )-adapted process such that for any bounded stopping
time T , E(XT ) ≤ E(AT ). Given t ≥ 0, set Vt = sup0≤s≤t Xs .
1. Show that for any stopping time T and any ǫ > 0, P (VT ≥ ǫ) ≤ 1ǫ E(AT ). (One can
use r τε = inf{t ≥ 0 : Xt ≥ ε} et Tn = T ∧ n ∧ τε .)
2. Show that for any stopping time T , and any δ > 0, ǫ > 0, si S = inf{t ≥ 0 : At ≥ δ},
P (VT ≥ ǫ, AT ≤ δ) ≤ P (VT ∧S ≥ ε) ≤ 1ǫ E(δ ∧ AT ).
3. Let F : [0, +∞[→ [0, +∞[ be a function differentiable on ]0, +∞[, strictly increasing,
′
such that F (0) = 0 and for any x > 0, u → F u(u) 1[x,+∞[(u) ∈ L1 (λ), where λ denotes
the Lebesgue measure. Let G :]0, +∞[→]0, +∞[ denote the function defined by
Z +∞ ′
F (u)
du.
G(x) = 2F (x) + x
u
x
R +∞ ′
(a) Prove that G′ (x) = F ′ (x) + x F u(u) du.
(b) Prove that
Z +∞
E(F (VT )) =
P (VT ≥ u)F ′(u)du
Z0 +∞
Z +∞
1
′
≤
E AT 1{AT ≤u} F ′ (u)du
2P (AT ≥ u)F (u)du +
u
0
0
≤ E(G(AT )).
Deduce that for every p ∈]0, 1[, every stopping time T , E(VTp ) ≤
2009-2010
2−p
E(ApT ).
1−p
Stochastic Calculus 2 - Annie Millet
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Stochastic differential equations
2 Stochastic differential equations
These processes, which generalize ordinary differential equations, are fundamental in
finance. They model the price of financial assets.
Let us recall that an ordinary differential equation (ODE) on [0, +∞[×R is as follows :
yt′ = f (t, yt) and y0 = y,
(2.1)
where y : [0, +∞[→ R is the unknown function and f : [0, +∞[×R → R is a given function.
The study of ODE is extremely important for applications of mathematics, for example in
physics. In general, it is impossible to give an explicit analytic expression of the solution to an
ODE. However, one wants to know whether such an equation has a unique solution (and ways
to approximate it numerically). Uniqueness is important to prove that the approximation
scheme converges to the (unique) solution. A basic existence and uniqueness criterion is the
following :
Theorem 2.1 (The Cauchy-Lipschitz theorem) Let K > 0 be a constant such that for all
t ∈ [0, +∞[, x, y ∈ R :
|f (t, x) − f (t, y)| ≤ K|x − y|
(global Lipschitz condition),
|f (t, x)| ≤ K(1 + |x|) (linear growth condition).
Then the ODE (2.1) has a unique solution defined on [0, +∞[.
The global Lipschitz property is natural if the solution is required to be defined on the entire
half-line [0, +∞[. Indeed, let y0 = 1 and f (t, y) = y 2 , then it is easy to see that the unique
solution to the corresponding ODE is (2.1) the function defined by yt = 1/(1 − t), and that
this solution « explodes » at t = 1.
A stochastic differential equation (SDE) is a perturbation of (2.1) by a stochastic term
which models a « noise » around the deterministic phenomenon described by (2.1). The
simplest perturbation consists in adding a Brownian motion. This models the fact that on
disjoint time intervals, the perturbation is the sum of a very large number of « small »
shocks, that is independent random variables which have the same distribution. Using the
central limit theorem, we see that, when properly renormalized, the distribution of this sum
is close to that of a Gaussian random variable. The number of small shocks is supposed to
be proportional to the length of the interval, which means that the variance is a multiple
of the length of this time interval. Thus, we consider the SDE dYt = f (t, Yt )dt + σdBt and
Y0 = y. This formulation is formal (the Brownian motions is a.s. nowhere differentiable) ;
the only rigorous formulation is in integrated form :
Z t
Yt = y +
f (s, Ys )ds + σBt
0
for all t ≥ 0. It is usual to denote solutions to SDE by capital letters and keep lower case
letters for solutions of ODEs.
The trajectory of the solution y to the ODE (2.1) is regular and deterministic. The
added Brownian motion implies that for « small σ », the trajectory of the solution to the
above SDE, which is a.s. non-differentiable, is close to that of (2.1), and oscillates around
it. However, for large σ the trajectories of the SDE are no longer similar to that of (2.1).
Stochastic Calculus 2 - Annie Millet
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25
Strong solution - Diffusion.
The next figure shows the trajectories on [0, 2] of the solution of the ODE
yt′ = −yt , et y0 = 5
(that is yt = 5e−t ) and of the SDEs
Yt = 5 −
Z
t
Ys ds + σBt
0
with σ = 0.4 and σ = 2 obtained by simulation.
6
5
4
3
2
1
0
−1
−2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.1 Strong solution - Diffusion.
In the sequel, we will study stochastic differential equations with the following precise
definition.
Fix a right-continuous filtration (Ft ) with complete σ-algebras (but where F0 need not
be the σ-algebra of null sets) and B a (Ft )-Brownian motions taking values in Rr , ξ a
Rd -valued random variable independent of FsB = σ(Bs , s ≥ 0) and Borel functions
σ : [0, +∞[×Rd → M(d, r) ∼ Rdr
and
b : [0, +∞[→ Rd .
The stochastic differential equation (SDE) with initial condition ξ, diffusion coefficient
σ and drift coefficient b is a process X such that for any t ≥ 0,
Z t
Z t
Xt = ξ +
σ(s, Xs )dBs +
b(s, Xs )ds.
(2.2)
0
0
Equation (2.2) will also be denoted as follows :
dXt = σ(t, Xt )dBt + b(s, Xs )ds,
X0 = ξ.
We at first prove an existence and uniqueness result in the strong sense.
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2
Stochastic differential equations
Definition 2.2 Let B be a given Brownian motion, ξ a random variable independent of the
σ-algebra σ(Bs , s ≥ 0) and let (Ft ) be the filtration defined as follows : for any t ≥ 0, Ft
is the completion of the σ-algebra σ(σ(Bs , 0 ≤ s ≤ t), σ(ξ)). A strong solution of (2.2) is a
process (Xt ) with a.s. continuous trajectories such that
(i) (Xt ) is adapted to the filtration (Ft ),
(ii) X0 = ξ a.s.,
Rt
P
(iii) For every i = 1, · · · , d, t ≥ 0 0 |bi (s, Xs )| + rk=1 |σki (s, Xs )|2 ds < +∞ a.s.
(iv) For every t ≥ 0 and every i = 1, · · · , d,
Xti
=
X0i
+
r Z
X
t
0
k=1
σki (s, Xs )dBsk
+
Z
t
bi (s, Xs )ds.
0
The following theorem is fundamental ; it gives sufficient conditions on the coefficients
(similar to that for ODEs) which ensure existence and uniqueness of a strong solution to
(2.2). Let |.| denote the Euclidian norm on Rd or on Rd×r . The following conditions extend
that used for ODEs.
Definition 2.3 Fix T > 0. A map ϕ : [0, T ] × Rd → RN satisfies
(i) The global Lipschitz condition on [0, T ] if there exists a constant C > 0 such that
|ϕ(t, x) − ϕ(t, y)| ≤ C|x − y| , ∀t ∈ [0, T ], ∀x, y ∈ Rd .
(2.3)
(ii) The global growth condition on [0, T ] if there exists a constant C > 0 such that
|ϕ(t, x)| ≤ C(1 + |x|) , ∀t ∈ [0, T ], ∀x, y ∈ Rd .
(2.4)
When ϕ is defined on [0, +∞[×Rd and conditions (2.3) and (2.4) are satisfied on any
interval [0, T ] with constants which do not depend on T , we say that ϕ satisfies the global
Lipschitz and growth conditions.
It is easy to see that when ϕ is globally Lipschitz, the growth condition (2.4) can be
deduced from
sup |ϕ(t, 0)| < +∞.
t≥0
The proof of the existence and uniqueness theorem relies on the following Gronwall lemma.
Lemma 2.4 (Gronwall’s lemma) Let f : [0, T ] → [0, M] be a non-negative bounded Borel
function, a and b ≥ 0 be real constants such that for any t ∈ [0, T ]
f (t) ≤ a +
Z
t
bf (s)ds.
0
Then for any t ∈ [0, T ], f (t) ≤ a ebt .
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Strong solution - Diffusion.
Proof. Iterate the upper estimate under the integral. This yields for all n ≥ 1
Z t
Z s
f (t) ≤ a + b
a + b
f (u) du ds
0
0
Z
2
f (u) du ds
≤ a + abt + b
0≤u≤s≤t
Z
Z u
2
a + b
f (v) dv du ds
≤ a + abt + b
0≤u≤s≤t
0
Z tZ sZ u
Z
3
2
du ds + b
≤ a + abt + ab
f (v) dv du ds
0≤u≤s≤t
n n
2 2
≤ a + abt + a
0
0
0
n+1 n+1
b t
b t
bt
+···+a
+M
,
2!
n!
(n + 1)!
where in the last integral we have upper estimated the function f by M. As n → ∞, the
last inequality yields f (t) ≤ a ebt .
2
Note that one can weaken the assumptions, requiring that the coefficients only satisfy a
« local » Lipschitz condition, but that they satisfy the global growth condition.
Definition 2.5 The function ϕ is locally Lipschitz on [0, T ] with respect to the space variable
x if for every n ≥ 1 there exists Kn > 0 such that for all t ∈ [0, T ] and x, y ∈ Rd :
|ϕ(t, x) − ϕ(t, y)| ≤ Kn |x − y|
if |x| ≤ n and |y| ≤ n.
(2.5)
If ϕ is defined on [0, +∞[ and is locally Lipschitz (that is satisfies (2.5) ) on every interval
[0, T ] with a sequence of constants Kn which does not depend on T , one says that it is locally
Lipschitz.
Theorem 2.6 (Strong existence and uniqueness theorem) Let σ and b be coefficients defined
on [0, +∞[ which satisfy the global Lipschitz and growth conditions (2.3) and (2.4). Then,
if B is a standard r-dimensional Brownian, for every square integrable random variable X0
independent of σ(Bs , s ≥ 0) the SDE
Xt = X 0 +
Z
t
σ(s, Xs )dBs +
0
Z
0
t
b(s, Xs )ds a.s. ,
∀t ≥ 0
(2.6)
has a unique strong solution. This strong solution X of (2.6), is adapted to the filtration (Ft )
where Ft is the completion of σ(X0 , σ(Bs , 0 ≤ s ≤ t)), and has a.s. continuous trajectories.
Furthermore, there exists a constant C̃(C, T ) such that foe every t ≥ 0 :
2
(2.7)
E sup |Xs | ≤ C̃(C, T )eC̃(C,T )t [1 + E(|X0 |2 )].
0≤s≤t
If the initial condition X0 ∈ Lp , 2 ≤ p < +∞, then there exists a constant C̄(C, T, p) such
that
p
(2.8)
E sup |Xt | < C̄(C, T, p)[1 + E(|X0 |p )].
0≤t≤T
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2
Stochastic differential equations
Proof. Fix T > 0 ; we prove the existence and uniqueness of a strong solution on the time
interval [0, T ].
Uniqueness Here we only suppose that condition (2.5) holds. Let X and X̄ be strong
solutions of (2.6) ; they have a.s. continuous trajectories. For any n ≥ 0, let τn = inf{t ≥ 0 :
|Xt | ≥ n}, τ̄n = inf{t ≥ 0 : |X̄t | ≥ n} and Tn = τn ∧ τ̄n . Then Tn → ∞ a.s. Furthermore,
Z t∧Tn
Z t∧Tn
b(s, Xs ) − b(s, X̄s ds.
σ(s, Xs ) − σ(s, X̄s ) dBs +
Xt∧Tn − X̄t∧Tn =
0
2
0
Pd Pr
i 2
k=1 |σk | .
The isometry of stochastic integrals
Given a d × r matrix σ, set kσk = i=1
and Schwarz’s inequality imply that for all t ∈ [0, T ],

2 
r Z t∧Tn
d X
X
i
i
k 
2

[σk (s, Xs ) − σk (s, X̄s )] dBs E(|Xt∧Tn −X̄t∧Tn | ) ≤ 2E
i=1 k=1 0
2 !
Z t∧Tn
+ 2E [b(s, Xs ) − b(s, X̄s )] ds
0
Z t∧Tn
Z t∧Tn
2
|b(s, Xs ) − b(s, X̄s )|2 ds
kσ(s, Xs ) − σ(s, X̄s )k ds + 2tE
≤ 2E
0
0
Z t
≤ 2(T + 1)Kn2
E |Xs∧Tn − X̄s∧Tn |2 ds.
0
Gronwall’s lemma applied to the continuous function f defined by f (t) = E(|Xt∧Tn −X̄t∧Tn |2 )
implies that the processes Xt∧Tn and X̄t∧Tn are indistinguishable. As n goes to infinity, we
deduce that X and X̄ are indistinguishable as well.
Existence In order to highlight the main ideas and to keep notations simple, we suppose
here that r = d = 1.
As for ODEs, we construct a solution by a Picard iteration scheme. Let X (n) be a
sequence of processes defined as follows.
(0)
Xt
(n+1)
Xt
= X0 , ∀t ∈ [0, T ],
Z t
Z t
(n)
= X0 +
σ(s, Xs )dBs +
b(s, Xs(n) )ds, ∀n ≥ 0, ∀t ∈ [0, T ].
0
(2.9)
0
(i) We prove by induction that for any integer n ≥ 0,
(n)
sup E(|Xt |2 ) < +∞.
(2.10)
0≤t≤T
Using the linear growth condition (2.4), this yields that the processes X (n) are well-defined
(n)
(n)
since b(t, Xt ) ∈ H1T (Ft ) and σ(t, Xt ) ∈ H2T (Ft ). By definition, X0 ∈ L2 and (2.10)
holds for n = 0. Suppose that (2.10) holds for n ; we prove it for n + 1. For any t ∈ [0, T ],
Schwarz’s inequality, the L2 isometry of stochastic integrals, condition (2.4) and Fubini’s
theorem imply that
Z t
Z t
(n+1) 2
2
(n) 2
(n) 2
E(|Xt
| ) ≤ 3 E(|X0 | ) + T
E(|b(s, Xs )| )ds +
E(|σ(s, Xs )| )ds
0
0
Z t
2
2
(n) 2
≤ 3 E(|X0 | ) + 3(T + 1) C
1 + E(|Xs | ds ,
0
Stochastic Calculus 2 - Annie Millet
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2.1
29
Strong solution - Diffusion.
which proves (2.10).
(ii) There exists a constant C > 0 such that
(n)
sup sup E(|Xt |2 ) ≤ C[1 + E(|X0 |2 )]eCt .
(2.11)
n≥1 t∈[0,T ]
(1)
Indeed, using part (i), we deduce that for any t ∈ [0, T ], E(|Xt |2 ) ≤ C[1 + E(|X0 |2 )], while
for n ≥ 1,
Z
(n+1) 2
E(|Xt
|)
t
2
≤ C[1 + E(|X0 | )] + C
0
E(|Xs(n) |2 )ds.
Iterating this inequality and using induction on n, we deduce that for any n ≥ 1 and
t ∈ [0, T ],
h
(Ct)2
(Ct)n i
| ) ≤ C[1 + E(|X0 |2 )] 1 + Ct +
,
+···+
2!
n!
(n+1) 2
E(|Xt
which concludes the proof of (2.11).
(iii) We next prove that the sequence (X (n) ) converges a.s. uniformly on the interval
[0, T ] to a continuous, (Ft )-adapted process X such that (2.7) is satisfied. For all n ≥ 1, set
(n+1)
Xt
(n)
− Xt
= Ant + Mtn ,
where
Ant
=
Z
t
0
b(s, Xs(n) )
−
b(s, Xs(n−1) )
ds ,
Mtn
=
Z
t
0
σ(s, Xs(n) ) − σ(s, Xs(n−1) ) dBs .
(n+1)
(n)
Then, we have E(sup0≤s≤t |Xs
− Xs |2 ) ≤ 2E sup0≤s≤t |Ans |2 + 2E sup0≤s≤t |Msn |2 .
Schwarz’s inequality and (2.3) imply that for any r ∈ [0, t],
Z t
Z t
n 2
(n)
(n−1) 2
|At | ≤ t
|b(s, Xs ) − b(s, Xs
)| ds ≤ CT
|Xs(n) − Xs(n−1) |2 ds
0
0
(n)
(n−1)
Furthermore, (2.11) shows that the process s → σ(s, Xs )−σ(s, Xs
) belongs to H2T (Ft ).
Indeed, (2.3) shows that
Z t
Z t
2
(n)
(n−1)
2
σ(s, X ) − σ(s, X
E
) ds ≤ C
E(|Xs(n) − Xs(n−1) |2 )ds
s
s
0
0
Z t
≤ 2C 2
E(|Xs(n) |2 ) + E(|Xs(n−1) |2 ds < +∞.
0
Applying Doob’s inequality to the continuous martingale (Mtn , t ≥ 0), and using (2.3) we
deduce that
Z t
n 2
E sup |Ms | ≤ 4
E(|σ(s, Xs(n) ) − σ(s, Xs(n−1) )|2 )ds
0≤s≤t
0
Z t
2
≤ 4C
E(|Xs(n) ) − Xs(n−1) )|2 )ds.
0
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Stochastic differential equations
Both inequalities imply that for L := L(T ) = 2C(T + 4C),
E
sup
0≤s≤t
|Xs(n+1)
Xs(n) |2
−
≤L
Z
t
E(|Xs(n) ) − Xs(n−1) )|2 ds,
0
and iterating this inequality, we deduce by induction on n that
(Lt)n
(1)
(n+1)
(n) 2
E sup |Xs
− Xs | ≤ C̄
où C̄ = sup E(|Xt − X0 |2 ) < +∞.
n!
0≤s≤t
0≤t≤T
n
(n+1)
(n)
The Markov inequality implies that if An = sup0≤s≤T |Xs
− Xs | ≥
2n
P (An ) ≤ 2 E
sup
0≤s≤T
|Xs(n+1)
−
Xs(n) |2
≤ C̄
1
2n
(4Lt)n
.
n!
o
, one has
The Borel Cantelli yields that P (lim sup An ) = 0, that is, for almost every ω, there exists an
(n+1)
(n)
integer N(ω) such that sup0≤s≤T |Xs
(ω)−Xs (ω)|2 ≤ 21n for every n ≥ N(ω). Therefore,
P
(n)
(0)
(k+1)
(k)
n−1
the sequence Xt (ω) = Xt (ω)+ k=0
Xt
(ω)−Xt (ω) converges uniformly to Xt (ω).
Set X = 0 on the null set where the sequence (X n (ω) does not converge. The process X is
(a.s.) continuous, (Ft ) adapted as a.s. limit of a sequence on (Ft )-adapted process (recall
that the σ-algebras Ft are complete). Finally,
E
(n)
|Xt |2
sup
0≤t≤T
2
≤ 2E(|X0 | ) + 2
n−1
X
≤ 2E(|X0 |2 ) + 2C̄
Fatou’s lemma yields E sup0≤t≤T |Xt |
2
E
k=0
n−1
X
k=0
sup
0≤t≤T
(k+1)
|Xt
−
(k)
Xt | 2
(4Lt)k
≤ 2E(|X0 |2 ) + 2C̄e4LT .
k!
≤ 2E(|X0 |2 ) + 2C̄e4LT , which proves (2.7).
(n)
(iv) Finally, we prove that for all t ∈ [0, T ], the sequence Xt converges in √
L2 to Xt , and
1
that X satisfies (2.6). For every x > 0, Stirling’s formula implies that n! ∼ 2πnn+ 2 e−n .
P∞ xk 21
≤ C(x)m−1 . For every t ∈ [0, T ] and n > m,
Hence for large enough n, k=m k!
(n)
kXt
−
(m)
Xt k 2
≤
n−1
X
k=m
+∞ X
(4Lt)k 2
C̄
≤
→0
k!
k=m
1
(k+1)
kXt
−
(k)
Xt k 2
(n)
as m → +∞. The sequence Xt is Cauchy in L2 ; it converges in L2 to a square integrable
random variable Yt . Therefore, it has an a.s. converging sub-sequence which converges a ;s.
to Yt and we have Yt = Xt a.s.
Furthermore, (2.7) and the growth condition imply that the processes s → b(s, Xs ) and
s → σ(s, Xs ) belong respectively to H1T (Fs ) and H2T (Fs ). Fubini’s theorem shows that
E
Z
0
T
(n)
|Xt
−
(m)
Xt |2 dt
≤ C̄
Z
0
T
+∞ X
(4Lt)k 2
1
k=m
Stochastic Calculus 2 - Annie Millet
k!
!2
dt → 0 when m → +∞.
2009-2010
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31
Strong solution - Diffusion.
Using Fatou’s lemma and the Lipschitz condition (2.3) we deduce that for every m,
E
Z
T
|Xt −
0
(m)
Xt |2 dt
≤ lim inf E
n
Z
T
0
(n)
|Xt
−
(m)
Xt |2 dt
→0
Rt
(n)
as m → ∞. Schwarz’s inequality and the Lipschitz condition (2.3) yield that 0 b(s, Xs )ds
R
t
converges in L2 to 0 b(s, Xs )ds, while the isometry of stochastic integrals shows that
Rt
Rt
(n)
2
σ(s,
X
)dB
converges
in
L
to
σ(s, Xs )dBs . We deduce that X satisfies (2.6) ; indeed,
s
s
0
0
one lets n → ∞ in (2.9) and identifies the L2 -limit of each hand-side in (2.9).
If the integrability assumption of the initial condition is stronger, the proof above can easily
be adapted, using the Burkholder-Davies-Gundy inequality instead of the L2 isometry of
stochastic integrals. (1.13) yields (2.8).
2
Example 2.7 (Geometric Brownian motion) Let σ and b be real numbers, S0 > 0 be
positive real. The geometric Brownian is the solution to the SDE
St = S0 +
Z
t
σSs dBs +
0
Z
t
bSs ds.
(2.12)
0
The above theorem 2.6 shows that this SDE has a unique strong solution since the coefficients
σ(t, x) = σx and b(t, x) = bx obviously satisfy the global Lipschitz and growth conditions.
For any t ≥ 0, set
σ2
t .
(2.13)
Xt = S0 exp σBt + b −
2
2
Apply Itô’s formula to the function f (t, x) = S0 exp[σx + (b − σ2 )t] and to the Brow2
2
(t, x) = (b − σ2 )f (t, x), ∂f
(t, x) = σf (t, x), ∂∂xf2 (t, x) = σ 2 f (t, x),
nian motion B. Then ∂f
∂t
∂x
f (t, Bt ) = Xt and f (0, B0 ) = S0 . This yields
Z t
Z
Z t
Z t
Z t
1 t 2
σ2
Xs ds +
Xt = S 0 +
σXs dBs +
σ Xs ds = S0 +
σXs dBs +
b−
bXs ds,
2
2 0
0
0
0
0
that is X is a solution of (2.12). The uniqueness of the strong solution implies that X = S.
We deduce that St > 0 for every t ≥ 0 a.s. Using again Itô’s formula with the function
g(x) = ln(x) and the process St , we deduce that if one requires the process S to take only
positive values (which we have just checked) we deduce that for Yt = ln(St ),
Yt = Y0 +
Z
t
0
Ss
σ dBs +
Ss
Z
0
t
1
Ss
b ds −
Ss
2
Z
t
0
σ 2 Ss2
σ2
ds = Y0 + σBt + b −
t,
Ss2
2
which means that the exponential form (2.13) of the solution is indeed « natural ». In
exercise 2.4, we will extend this observation to a more general linear SDE. Furthermore, the
2
distribution of ln(St )−ln(S0 ) is Gaussian N ((b− σ2 )t, σ 2 t), which means that the distribution
of St is log-normal. The figures below shows simulations of a geometric Brownian motion
on [0, 1], with σ = 0.5 (resp. σ = 2), b = 2 and S0 = 1.
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Stochastic differential equations
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
−1
−2
0.0
2.2 Weak solution.
In many applications, it is more natural to give in the model the coefficients b(t, x) and
σ(t, x), as well as the distribution of the initial condition X0 , but neither the probability
space not the Brownian motion B. One looks then for the pair (B, X) satisfying (2.6), which
give the following notion.
Definition 2.8 A weak solution of equation
dXt = σ(t, Xt )dBt + b(t, Xt )dt
(2.14)
is a triplet (Ω, F , (Ft), P ), B, X where
(i) (Ω, F , P ) is a probability space endowed with a complete right-continuous filtration
(Ft ).
(ii) (Bt ) is a r-dimensional (Ft )-Brownian motion and (Xt ) is a (Ft )-adapted, d-dimensional
continuous process such that the properties (iii) and (iv) in definition 2.2 are satisfied.
The distribution of X0 is called the initial distribution. One no longer requires the filtration (Ft ) to be the smallest one for which B and X0 are measurable, and the solution Xt
at time t is no longer a measurable function of the Brownian trajectory (Bs , s ≤ t) and of
X0 . However, since B is a (Ft )-Brownian motion and X is (Ft )-adapted, Xt is independent
of the trajectory of B after time t, that is of the increments (Br − Bt , r ≥ t).
A strong solution is clearly a weak one but the converse is false as shown in the following
example. The existence of weak solutions can indeed be shown under weaker assumptions
on the coefficients.
Stochastic Calculus 2 - Annie Millet
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33
Some properties of diffusions
Example 2.9 (The Tanaka equation) Let sign(x) = 1 if x ≥ 0 and sign(x) = −1 if x < 0.
The SDE
Z t
Xt =
sign(Xs )dBs
(2.15)
0
has a weak solution, but does not have a strong solution.
Let Ω, F , (Ft), B, X be a weak solution. Then the process X is a continuous martingale
Rt
with quadratic variation hX, Xit = 0 sign(Xs )2 ds = t. Using P. Lévy’s characterization of
the Brownian motion (Theorem 1.33), X is a (Ft )-Brownian. This immediately yields the
« construction » of a weak solution.
LetR (Xt ) be a Brownian motion and (Ft ) denote its natural filtration. For any t ≥ 0 set
t
Bt = 0 sign(Xs )dXs . Then (Bt ) is a continuous (Ft )-martingale with quadratic variation
hB, Bit = t. P. Lévy’s characterization of the Brownian motion proves that B is an (Ft )Brownian. Furthermore, dBt = sign(Xt )dXt , that means dXt = sign(Xt )dBt .
Thus, on the probability space (Ω, F , (Ft , P ), the pair (B, X) is a weak solution to the
Tanaka equation (2.15). It is easy to see that the pair (B, −X) is also a weak solution to
(2.15). This shows that this equation does not have a unique « pathwise » solution, but at
most a unique solution « in distribution ».
The lack of strong solution will not be proved here.
Definition 2.10 We say that uniqueness
in distribution of the solution
(2.14) holds if two
weak solutions (Ω, F , (Ft), P ), B, X and (Ω̃, F̃, (F̃t ), P̃ ), B̃, X̃ have the same distribution, that is given any n ≥ 1 and times 0 ≤ t1 < t2 < · · · < tn , the vectors (Xt1 , · · · , Xtn )
and (X̃t1 , · · · , X̃tn ) have the same distribution.
The above argument shows that there is uniqueness in distribution for the Tanaka equation
(2.15). Indeed, we have proved that any weak solution is a Brownian motion.
Notice that when the coefficients σ and b satisfy the conditions (2.3) and (2.4), Theorem
2.6 proves existence and uniqueness of a strong solution to (2.6). There is also uniqueness of the weak solution
and uniqueness in distribution
for equation (2.14). Indeed, let
(Ω, F , (Ft), P ), B, X and (Ω̃, F̃ , (F̃t ), P̃ ), B̃, X̃ be two weak solutions and let Y and Ỹ
be two continuous strong
solutions of (2.6) defined
respectively on the probability spaces
(Ω, F , (Ft), P ), B, X and (Ω̃, F̃ , (F̃t ), P̃ ), B̃, X̃ . Then Theorem 2.6 implies that Xt = Yt
holds for all t ≥ 0 P a.s. (i.e., that using the continuity of the trajectories of both processes,
one obtains a null set built on rationals such that on its complement the trajectories of X
and Y agree). Similarly, X̃t = Ỹt for all t ≥ 0 P̃ a.s. To prove weak uniqueness, it is enough
to check that the distributions of X and X̃ are the same. Using the Picard iteration scheme
(2.9) for X and X̃, one checks that for every n ≥ 0, the distributions of the continuous
(n)
(n)
processes (Bt , Xt ) and (B̃t , X̃t ) are the same.
2.3 Some properties of diffusions
2.3.1 Stochastic flows and the Markov property
Let σ and b be coefficients which satisfy the global Lipschitz conditions (2.3) and growth
conditions (2.4). The proof of Theorem 2.6 shows that for any s ≥ 0 and for any x ∈ Rd ,
the exists a unique strong solution (Xts,x , t ≥ s) of the SDE (2.6) starting from state x at
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Stochastic differential equations
time s, that is defined for t ≥ s by
Z t
Z t
s,x
s,x
Xt = x +
σ(u, Xu )dBu +
b(u, Xu )du.
s
(2.16)
s
Then X 0,x = X(x) is the strong solution to (2.6) with X0 = x, that is
Z t
Z t
Xt (x) = x +
σ(s, Xs (x))dBs +
b(s, Xs (x))ds
0
(2.17)
0
and we have shown that E(sup0≤t≤T |Xt (x)|2 ) < +∞.
Let us first check the following flow property.
s,Xs (x)
Theorem 2.11 (Stochastic flow property) For all 0 ≤ s ≤ t, Xt (x) = Xt
a.s.
Proof. Let t ≥ s ; then using (2.17) we deduce :
Z t
Z t
Xt (x) = Xs (x) +
σ(u, Xu (x))dBu +
b(u, Xu (x))du,
s
s
while (2.16) implies :
s,X (x)
Xt s
= Xs (x) +
Z
t
s
σ(u, Xus,Xs(x) )dBu
+
Z
t
s
b(u, Xus,Xs(x) )du.
s,X (x)
This shows that the processes (Xt (x), t ≥ s) and (Xt s , t ≥ s) are indistinguables since
they are both strong solutions of the same SDE with well behaved coefficients.
2
This stochastic flow property implies the « weak » Markov property.
Theorem 2.12 (The Markov property) Let (Xt (x), t ≥ 0) be the strong solution to the SDE
(2.6) with X0 = x. Then (Xt (x), t ≥ 0) is a Markov process for the natural filtration (FtB ) of
the Brownian motion. More precisely, for any function positive (or bounded) Borel function
f : Rd → R, 0 ≤ s, t,
Z
B
E[f (Xs+t (x))|Fs ] = E[f (Xs+t (x))|Xs (x)] = Φt (Xs (x)) = Ps,s+t(Xs (x), dz) f (z),
s,y
s,y
where Φt (y) = E[f (Xs+t
)] and Ps,s+t (y, dz) is the distribution of Xs+t
.
s,y
Proof. The process (Xs+t
, t ≥ 0) is the unique strong solution to the SDE (2.6) with initial
condition y, coefficients (t, x) → σ(s + t, x) and (t, x) → b(s + t, x) and for the Brownian
motion B̃ defined by B̃t = Bs+t − Bs . It is a measurable function Ψ(y, B̃) and is hence
s,X (x)
independent of FsB = σ(Bu , 0 ≤ u ≤ s). This yields that Xs+t (x) = Xs+t s is equal to
Ψ(Xs (x), B̃). Since Xs (x) and B̃ are independent, Lemma 1.37 implies that for any bounded measurable function f , E(f (Xs+t(x)|FsB ) = Φt (Xs (x)) where Φt (y) = E[f ◦ Ψ(y, B̃)] =
s,y
)]. When conditioning with respect to Xs (x), a similar result based on the indeE[f (Xs+t
pendence of B̃ and Xs (x) proves that E(f (Xs+t (x))|Xs (x)) = Φt (Xs (x)) ;
2
When the coefficients are homogeneous, that is do not depend on time, the Markov property can be rewritten in a more precise (homogeneous) way and the strong Markov property
holds. Indeed, in that case, since the process defined by B̃t = Bs+t − Bs is a (Fs+t, t ≥ 0)s,y
and Xt (y) have the same
Brownian motion, we deduce that the random variables Xs+t
distribution and that the function Φt (y) used in the Markov property can be expressed as
s,y
Φt (y) = E[f (Xs+t
)] = E[f (Xt (y))] = Pt f (y). Furthermore, one has the following
Stochastic Calculus 2 - Annie Millet
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Some properties of diffusions
Theorem 2.13 (Homogeneous Markov property - Strong Markov property)
Let σki : Rd → R and bi : Rd → R, 1 ≤ i ≤ d, 1 ≤ k ≤ r be time-independent coefficients.
Suppose that there exists C > 0 such that for any i = 1, · · · , d and k = 1, · · · , r :
|σki (x) − σki (y)| + |bi (x) − bi (y)| ≤ C|x − y|,
Let X(x) denote the solution to the homogeneous SDE
Z t
Z t
Xt = X 0 +
σ(Xs )dBs +
b(Xs )ds
0
(2.18)
(2.19)
0
with initial condition X0 = x. This is a homogeneous strong Markov process, that is for every
a.s. finite (FtB )-stopping time τ and any positive (or bounded) Borel function f : Rd → R,
E[f (Xτ +t (x))|Fτ ] = E[f (Xτ +t (x))|Xτ (x)] = Pt f (Xτ (x)),
with Pt f (y) = E[f (Xt (y))].
Notice that in the time-homogeneous case, the global Lipschitz condition (2.18) immediately
yields a growth property
|σki (x)| + |bi (x)| ≤ C(1 + |x|).
(2.20)
In the general (non-homogeneous) case, one shows in a similar way that the space-time
process is a strong Markov one, that is for a positive Borel function f : [0, +∞[×Rd → R
and an a.s. finite stopping time τ , if
Φt (s, x) = E[f (s + t, Xts,x )]
then :
E[f (τ + t, Xτ +t (x))|Fτ ] = E[f (τ + t, Xτ +t (x))|Xτ (x)] = Φt (τ, Xτ (x)).
2.3.2 Infinitesimal generator
Convention : In this section, we consider coefficients σ and b which satisfy the conditions
(2.3) and (2.4) in the general case, or conditions of Theorem 2.13 in the time-homogeneous
case with a constant initial condition X0 .
Let C n,p ([0, +∞[×Rd ) denote the set of functions u : [0, +∞[×Rd → R of class C n with
respect to the time-variable t ∈ [0, ∞[ and of class C p with respect to the space variable
x ∈ Rd .
Definition 2.14 When X is the solution to the homogeneous equation (2.19), if one sets
a = σσ ∗ , the infinitesimal generator of X is the differential operator A defined for u ∈ C 2 (Rd )
by
d
d
X
∂2u
1 X i,j
∂u
i
a (x)
(x) +
(x) .
(2.21)
Au(x) =
b (x)
∂xi
2 i,j=1
∂xi ∂xj
i=1
For example, the infinitesimal generator of the Brownian motion on Rr (which is the
particular case d = r, σki (x) = δi,k and hence a(x) = Idr ) is 21 ∆, where ∆ is the Laplace operator defined on C 2 (RRr ). The infinitesimal
generator of a (real-valued) geometrical
Rt
t
Brownian motion Xt = x + 0 σXs dBs + 0 bXs ds is the operator defined on C 2 (R) by
∂2
∂
+ 12 σ 2 x2 ∂x
A = b x ∂x
2.
When the diffusion is not homogeneous, its infinitesimal generator depends on time.
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Stochastic Calculus 2 - Annie Millet
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Stochastic differential equations
Definition 2.15 Let a(t, x)
= σ(t, x) σ ∗ (t, x) denote the symmetric non negative matrix
P
r
d×d defined by ai,j (t, x) = k=1 σki (t, x) σkj (t, x). The infinitesimal generator of the diffusion
(Xt , t ≥ 0) solution to (2.6) is the differential operator defined on C 1,2 ([0, +∞[×Rd ) by :
At u(t, x) =
d
X
i=1
d
∂u
1 X i,j
∂2u
b (t, x)
(t, x) +
a (t, x)
(t, x) .
∂xi
2 i,j=1
∂xi ∂xj
i
(2.22)
Using Itô’s formula, we deduce that if f : [0, +∞[×Rd → R is of class C 1,2 and if X is a
solution to (2.6), one has
∂f
∂f
df (t, Xt ) = σ(t, Xt ) (t, Xt )dBt +
(t, Xt ) + At f (t, Xt ) dt.
(2.23)
∂x
∂t
This yields the following
Theorem 2.16 Let σ and b satisfy the global Lipschitz and growth conditions (2.3) and
(2.4), X0 ∈ Rd and X denote the strong solution to (2.6).
∂f
For any function f : [0, +∞[×Rd → R of class C 1,2 such that ∂x
is bounded for any
i
i = 1, · · · , d, the process (Mt ) defined by
Z t
∂f
Mt = f (t, Xt ) −
+ As f (s, Xs ) ds
(2.24)
∂s
0
is a (FtB )-martingale and for any a.s. bounded stopping time τ , the Dynkin formula is true :
Z τ ∂f
+ As f (s, Xs )ds .
E[f (τ, Xτ )] = f (0, X0) + E
∂s
0
Proof. The martingale property of M follows immediately from (2.23), Theorem 1.13, (2.7),
and from the growth condition on σ. These properties imply that s → σ(s, Xs ) ∂f
(s, Xs ) ∈
∂x
H2 .
The optional stopping theorem 1.7 shows that if τ is a.s. bounded, since the σ-field F0B
is that of null sets, one has M0 = E(Mτ ). The partial derivatives of f with respect to x are
bounded, so that |f (s, Xs ) − f (s, X0 )| is dominated by a linear combination of components
Xsi . Due to (2.7), for every K > 0, the random variable sups≤K |f (s, Xs ) − f (s, X0 )| is
square integrable. Furthermore, f is continuous and hence sups≤K |f (s, X0 )| < +∞. For any
bounded stopping time τ , the random variable f (τ, Xτ ) is integrable, which concludes the
proof.
2
Using (2.8), we may weaken the assumptions on the partial derivatives of f made in the
∂f
(t, x) has a polynomial growth in x uniformly in t, that is if
previous theorem. Indeed, if ∂x
i
there exists an exponent ai > 0 such that
∂f
(t, x) ≤ C(1 + |x|ai ),
sup ∂xi
t
then the growth condition on the coefficients σki (t, x) and the fact that the initial condition
is constant (hence in all Lp spaces, 1 ≤ p < +∞) imply that for all k, there exists a constant
C and an exponent bi,k > 0 such that for all t > 0
2
Z t
2
i
∂f
b
(s, Xs ) σk (s, Xs ) ds ≤ C 1 + sup E(|Xs | i,k ) < +∞.
E
0≤s≤t
0 ∂xi
Stochastic Calculus 2 - Annie Millet
2009-2010
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Some properties of diffusions
∂f
(t, x) have a polynomial growth, we may conclude that
Thus, if all partial derivatives ∂x
i
the stochastic integrals appearing in the Itô formula for f (t, Xt ) are continuous (square
integrable) martingales.
Using Theorem 2.16 with d = r = 1, we deduce immediately that if f ∈ C 1,2 has a partial
derivative ∂f
which is bounded (or with polynomial growth) and if
∂x
∂f
∂f
1
∂2f
∂f
(t, x) + At f (t, x) =
(t, x) + b(t, x) (t, x) + σ(t, x)2 2 (t, x) = 0,
∂t
∂t
∂x
2
∂x
then f (t, Xt ) is a ”true” (FtB )- martingale Similarly, for any r ∈ R,
Z t
h ∂f
i
r
−rt
Mt (f ) = e f (t, Xt ) − f (0, X0) −
e−rs
+ As f − rf (s, Xs ) ds
∂s
0
is a (FtB ) martingale. Thus we deduce that if
∂f
(t, x) + At f (t, x) − rf (t, x) = 0,
∂t
then the process e−rt f (t, Xt ) is an (FtB )-martingale. The coefficient r is called the discount coefficient and e−rt the discount factor. Under this condition, if f (T, x) = h(x), then
f (t, Xt ) = er(t−T ) E[h(XT )|FtB ]. This will be useful in finance, as we will see it later on. This
martingale property can be generalized to a non constant discount coefficient.
Theorem 2.17 Let σ and b satisfy the global Lipschitz and growth conditions (2.3) and
(2.4), let B be a standard r-dimensional Brownian motion, Xt be the solution to the SDE :
Z t
Z t
Xt = x +
σ(s, Xs ) dBs +
b(s, Xs ) ds
(2.25)
0
0
and At be its infinitesimal generator. Then for any lower-bounded continuous function ρ :
[0, T ] × Rd → [m, +∞[ and any function f : [0, T ] × Rd → Rd of class C 1 in t and of class
∂f
C 2 in x such hat the partial derivatives ∂x
of f have polynomial growth, then the process
i
Rt
Mtρ (f ) = e− 0 ρ(s,Xs ) ds f (t, Xt ) − f (0, X0 )
Z t R
∂f
− 0s ρ(u,Xu ) du
−
e
+ As f − ρ f (s, Xs ) ds
∂s
0
is a martingale for the filtration FtB ).
Proof : The Itô formula for a product yields
Rt
Rt
− 0 ρ(s,Xs ) ds
f (t, Xt ) = −ρ(t, Xt ) e− 0 ρ(s,Xs ) ds f (t, Xt ) dt
d e
Rt
+e− 0 ρ(s,Xs ) ds d f (t, Xt ) .
Itô’s formula for f (t, Xt ) yields :
e−
2009-2010
Rt
0
ρ(s,Xs ) ds
f (t, Xt ) = f (0, X0 )
Z t R
∂f
− 0s ρ(u,Xu ) du
+
e
+ As f − ρ f (s, Xs ) ds
∂s
0
d X
r Z t
Rs
X
∂f
(s, Xs ) σki (s, Xs ) dBsk .
e− 0 ρ(u,Xu ) du
+
∂x
i
i=1 k=1 0
Stochastic Calculus 2 - Annie Millet
38
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Stochastic differential equations
ρ isR lower bounded,
as in the previous theorem that the processes
RSince
∂f one proves
t
s
i
exp − 0 ρ(u, Xu ) du ∂xi (s, Xs ) σk (s, Xs ) dBsk are martingales, which concludes the proof.
0
2
In the case of homogeneous coefficients, the differential operator defined by (2.21) is that
of the homogeneous Markov process.
Theorem 2.18 The infinitesimal generator A of the homogeneous diffusion X(x) solution
to (2.19) with initial condition x is is the generator of the homogeneous Markov process,
that is : for every function f of class C 2 with bounded partial derivatives or order 1 et 2,
1
1
Af (x) = lim [Pt f (x) − f (x)] = lim [E(f (Xt (x)) − f (x)].
t→0 t
t→0 t
Proof. Using (2.16), we have for all t ≥ 0,
E[f (Xt (x))] = f (x) +
Z
0
t
E Af (Xs (x) ds.
Since the trajectories
of X are a.s. continuous, that of Af (Xs ) are continuous too. This
Rt
implies that t → 0 Af (Xs (x))ds is P a.s. differentiable at 0 with almost sure derivative
equal to Af (X0 (x)) = Af (x).
Furthermore, the partial derivatives of f are bounded by some constant r K > 0, so that
the linear growth condition imposed on the coefficients implies that
X
X
K|σki (x)σkj (x)| ≤ C(1 + |x|2 ).
|Af (x)| ≤ K
|bi (x)| +
i
i,j,k
The property (2.7) implies that for all t > 0,
2
sup |Af (Xs )| ≤ C 1 + sup |Xs | ∈ L1 .
0≤s≤t
0≤s≤t
Taking derivative under the expectation concludes the proof.
2
2.3.3 Comparison Theorem
The following allows to compare a.s. solutions to one-dimensional SDEs which have the
same diffusion coefficient when the drift coefficients and the initial conditions satisfy similar
inequalities. It is very useful. The proof, which uses arguments similar to that of strong
uniqueness, can be read in [7].
Theorem 2.19 (Comparison Theorem) Let (Bt , t ≥ 0) be a real Brownian motion, b1 , b2
and σ be globally Lipschitz functions from R to R, x1 ≥ x2 be real numbers. For i = 1, 2,
consider the SDE
Z t
Z t
i
i
Xt = xi +
bi (Xs ) ds +
σ(Xsi ) dBs
0
0
Suppose that b1 (x) ≥ b2 (x) for every x ∈ R. Then Xt1 ≥ Xt2 a.s. for all t ≥ 0.
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39
Bessel Processes
2.4 Bessel Processes
The strong existence and uniqueness theorem for the solution to SDEs under the classical
(global Lipschitz and growth) conditions on the coefficients cannot be applied in some cases
for the diffusion coefficient. It can be improved in the real case.
Theorem 2.20 (Yamada-Watanabe) Let d = r = 1. Suppose that b and σ have linear
growth, that is the global growth condition (2.4) is satisfied, that b satisfies the global Lipschitz
condition (2.3) and that
|σ(t, x) − σ(t, y)| ≤ ρ(|x − y|) , ∀t ≥ 0, ∀x, y ∈ R,
where ρ :]0, +∞[→]0, +∞[ is a strictly increasing Borel function such that ρ(0) = 0 and
Z ε
dz
= +∞
2
0 ρ (z)
for any ε > 0. Then if X0 = x ∈ R, equation (2.6) has at most one strong solution.
Rε
Clearly if σ is Lipschitz, the above property holds with ρ(x) = Cx and condition 0 dz
=
z2
+∞ is satisfied. The following very important class of SDEs can be proven to have a unique
strong solution by means of the previous theorem
Z tp
Xs dBs + δ t
(2.26)
Xt = x + 2
0
p
√ √
where B is a one-dimensional Brownian motions and x, δ > 0.√
Indeed, | x− y| ≤ |x − y|
and using the Yamada-Watanabe with the function ρ(x) = x, we obtain that (2.26) has
a unique strong solution. Furthermore, one can prove that this solution remains positive
(if x > √
0) and defined on [0, +∞[. However, Theorem 2.6 does not imply this result since
σ(x) = x is not Lipschitz at zero. The process X is called the square of a δ dimensional
Bessel process.
Let us connect square Bessel processes and the Brownian motion. Let n > 1 and let B =
(B 1 , B 2 , . . . , B n ) be a n-dimensional
motion. Let X be the process defined by
Pn Brownian
i 2
(B
)
>
0
a.s. for t > 0 and Itô’s formula implies
Xt = P
||Bt || ; then Vt = Xt2 =
t
i=1
n
i
i
dVt = i=1 2Bt dBt + n dt.
Let (x, y) denote the scalar product of the vectors x and y, and β be the process defined by
n
1 X i i
1
B dB , β0 = 0,
(Bt , dBt ) =
dβt =
Xt
||Bt || i=1 t t
This is a continuous square integrable martingale and Itô’s formula shows that (βt2 −t, t ≥ 0)
is a martingale. Paul Lévy’s characterization proves that β is a y definition, one has
p
dVt = 2Xt dβt + n dt = 2 Vt dβt + ndt .
Thus, Vt = kBt k2 is solution of a SDE of the form (2.26) with δ = n. Since Xt > 0 a.s.,
applying once more Itô’s formula yields
n − 1 dt
dXt = dβt +
2 Xt
where β is a Brownian motion.
X is called a Bessel process (BES) of dimension n, and V the square of a Bessel process
(BESQ). The Bessel processes will be generalized in the last chapter (this is the CoxIngersoll-Ross model).
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Stochastic differential equations
2.5 Diffusions and PDEs
2.5.1 Parabolic problem
Let I be an interval included in [0, +∞[. Let Kn,p (I × Rd ) denote the set of functions
u : I × Rd → R which are of class C n with respect to the time variable t ∈ I and of class
C p with respect to the space variable x ∈ Rd , and such that the partial derivatives have
polynomial growth, that means there exist constants C > 0 and integers k such that all the
partial derivatives v of u satisfy the inequality supt∈I |v(t, x)| ≤ C (1 + |x|k ). Let Kp (Rd )
denote the set of functions u : Rd → R of class C p with respect to x ∈ Rd such that all
partial derivatives have growth.
The simplest example of parabolic partial differential equation (PDE) connected to to
stochastic process is the heat equation in dimension 1, that is
∂u
σ2 ∂2 u
(t, x) =
(t, x) , (t, x) ∈]0, +∞[×R ,
∂t
2 ∂x2
(2.27)
where σ > 0 and the initial condition u(0, x) = f (x) is given by a Borel function f : R →
R with polynomial growth. For t > 0, let p(t; x, .) denote the density of x + σ Bt where
(Bt , t ≥ 0) is a standard one-dimensional Brownian motion, that is the function
p(t; x, y) =
2
(x−y)2
1
√
e− 2 σ2 t .
σ 2πt
2
∂ p
= σ2 ∂x
An easy computation shows that ∂p
2 and that for any pair of integers n, m, there
∂t
exist exponents k > 0 and l > 0 such that for t > 0, x, y ∈ R,
n+m
∂
|x−y|2
≤ C|x − y|k t−l e− 2σ2 t .
p(t;
x,
y)
∂tm ∂xn
Set
u(t, x) = E f (x + σ Bt )
R +∞
for any t ≥ 0 ; then u(0, x) = f (x). Furthermore, given t > 0, u(t, x) = −∞ p(t; x, y) f (y) dy
and the polynomial growth of f implies that one may derive with respect to t ∈]0, +∞[
and to x ∈ R under the integral. Indeed, it suffices to prove that for any integer p ≥ 2,
E sups≤t |Bs |p < +∞, which is a straightforward consequence of Doob’s inequality and
from the fact that Bt has a Gaussian distribution, so that E|Bt |p < +∞. Hence for all
positive integers m and n it holds for (t, x) ∈]0, +∞[×R
Z +∞ n+m
∂ n+m
∂
u(t, x) =
p(t; x, y) f (y) dy ;
m
n
m
n
∂t ∂x
−∞ ∂t ∂x
2
2
(t, x) = σ2 ∂∂xu2 (t, x)
the function defined by u(t, x) = E[f (x + σ Bt )] satisfies the PDE ∂u
∂t
on ]0, +∞[×R and belongs to K1,2 ([ε, T ] × R) pour tout ε > 0. One still has to check the
asymptotic behavior of u(t, y) as (t, y) → (0, x). Again, since f has polynomial growth, if
this function is continuous, the dominated convergence theorem implies that for x ∈ R,
lim(t,y)→(0,x) u(t, y) = f (x) .
Thus, we have checked that equation (2.27), with initial condition a continuous function
f with polynomial growth, has a solution. We now check that given the initial condition,
(2.27) has a unique condition, which will give a probabilistic interpretation to this PDE
Again, one can prove uniqueness by probabilistic arguments.
Stochastic Calculus 2 - Annie Millet
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Diffusions and PDEs
Theorem 2.21 Let u be a function of class C 1,2 on ]0, T ]×R which satisfies the heat equation
(2.27), such that sup |u(t, x)| has polynomial growth with respect to x and is such that for
0<t≤T
any x ∈ R,
lim
(t,y)→(0,x)
u(t, y) = 0. Then u(t, x) = 0 sur ]0, T ] × R.
One can consider similar problems to (2.27) on the time interval [0, T ] « reversing time »
and imposing a terminal condition which is a continuous function f :
∂v
2 ∂2v
(t, x) + σ2 ∂x
for (t, x) ∈ [0, T [×R .
2 (t, x) = 0
∂t
(2.28)
v(T, x) = f (x)
for x ∈ R ,
A solution v continuous on [0, T ] × R, such that v ∈ C 1,2 ([0, T [×R) ∩ K1,2 ([0, T − ε] × Rd )
for all ε > 0 is the following :
v(t, x) = E f (x + σBT −t ) = E f (x + σ(BT − Bt )) for (t, x) ∈ [0, T ] × R .
These properties can be extended from the Brownian motion B to diffusions, which gives
a probabilistic interpretation of the parabolic PDE defined in terms of the infinitesimal
generator At defined by (2.22).
Theorem 2.22 Let σ and b be coefficients which satisfy the conditions (2.3) and (2.4), f
a continuous function and (Xts,x , t ≥ s) the solution to (2.16).
A function v continuous on [0, T ] × Rd such that v ∈ K1,2 ([0, T [×Rd ) is solution to the
Cauchy problem if :
∂v
(t, x) + At v(t, x) = 0 for (t, x) ∈ [0, T [×Rd ,
∂t
(2.29)
v(T, x) = f (x)
for x ∈ Rd ,
The function v has the following probabilistic representation :
v(t, x) = E f (XTt,x )
(2.30)
2.5.2 The Feynman-Kac formula
Let us study a PDE more general than that in the Cauchy problem (2.29) introduced in
the above section. Let σ and b be coefficients which satisfy the global Lipschitz and growth
conditions (2.3) and (2.4). For t ∈ [0, T [, let Xst,x denote the process solution to
Z s
Z s
t,x
t,x
Xs = x +
σ(u, Xu )dBu +
b(u, Xut,x )du , ∀s ∈ [t, T ].
(2.31)
t
t
Definition 2.23 Let At be the infinitesimal generator defined K1,2 ([0, T ]×Rd ) by (2.22). Let
m denote a real number, f : Rd → R, g : [0, T ] × Rd → Rd and ρ : [0, T ] × Rd → [m, +∞[
be continuous functions. the function v ∈ K1,2 ([0, T [×Rd ) satisfies the Cauchy problem
with differential operator At , potential ρ, Lagrangian g and terminal condition f if it is a
continuous function on [0, T ] × Rd such that
∂v
(t, x) + At v(t, x) − ρ(t, x) v(t, x) + g(t, x) = 0 for (t, x) ∈ [0, T [×Rd ,
∂t
(2.32)
v(T, x) = f (x) for x ∈ Rd .
Let us at first consider the case g = 0, that is only the situation where there is a
discounting function ρ.
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Theorem 2.24 (Feynman-Kac’s Theorem) Let σ and b be functions which satisfy the
global de Lipschitz and growth conditions (2.3) and (2.4), f, ρ be functions which satisfy the
assumptions of definition 2.23. Suppose that f ∈ K0 (Rd ). For t ∈ [0, T [ and x ∈ Rd , let
(Xst,x , s ∈ [t, T ]) denote the process solution to (2.31) and (Ft = FtB ) denote the natural
filtration defined by the Brownian motion B. Then a solution v to the Cauchy problem
∂v
(t, x) + At v(t, x) − ρ(t, x) v(t, x) = 0 for (t, x) ∈ [0, T [×Rd ,
∂t
v(T, x) = f (x) for x ∈ Rd .
has the following stochastic representation :
h
i
RT
t,x
v(t, x) = E f (XTt,x )e− t ρ(s,Xs ) ds .
(2.33)
Furthermore, if (Xt = Xt0,x , t ∈ [0, T ]) is the diffusion solution to (2.2), then for every
t ∈ [0, T ],
i
h RT
(2.34)
v(t, Xt ) = E e− t ρ(s,Xs ) ds f (XT )Ft .
Proof. An immediate generalization of (2.8) based on the Gronwall lemma and on Burkholder’s inequality proves that for any p ∈ [1, +∞[, E(supt≤s≤T |Xst,x |p ) ≤ Cp (1 + |x|p ).
The polynomial growth of the partial derivatives of v implies that for any i ≤ d, k ≤ r,
∂v
s ∈ [t, T ], if one sets Ys = ∂x
(s, Xst,x ) σki (s, Xst,x ), one has |Ys | ≤ C(1 + |Xst,x |p ), and hence
i
Rs
RT
t,x
E t |Ys |2p ds < +∞. Furthermore, the lower estimate of ρ proves that e− t ρ(u,Xu )du is
bounded by e|m|(T −t) . Using Itô’s formula (as in the proof of Theorem 2.17), we deduce that
if v is a solution to the Cauchy problem (2.32) with g = 0, the process
Rs
t,x
Mst,x = e− t ρ(u,Xu ) du v(s, Xst,x ) , s ∈ [t, T [
is a (Fs , s ∈ [t, T [)-martingale, which implies that for any t ≤ s < T , Mtt,x = E Mst,x | Ft .
For any integer n ≥ 1, let τn = inf{s ≥ t , |Xst,x | ≥ n}. The optional sampling theorem for
martingales implies that for s ∈ [t, T [ and any n ≥ 1,
t,x
t,x
),
|Ft) = E(Ms∧τ
Mtt,x = E(Ms∧τ
n
n
which implies that
R s∧τn
t,x
t,x
v(t, x) = E e− t ρ(u,Xu )du v(s ∧ τn , Xs∧τ
) .
n
In order to use the terminal condition and deduce (2.33), one lets s converge to T and n go
to +∞. The a.s. continuity of the trajectories of X.t,x the continuity property of v and of ρ
on [0, T ] × Rd prove that
e−
R s∧τn
t
ρ(u,Xut,x )du
t,x
) → e−
v(s ∧ τn , Xs∧τ
n
R T ∧τn
t
ρ(u,Xut,x )du
v(T ∧ τn , XTt,x∧τn ) a.s. when s → T.
Furthermore, the definition of τn proves that there exists a constant C(n) such that
R s∧τn
t,x
t,x
sup e− t ρ(u,Xu )du v(s ∧ τn , Xs∧τ
)| ≤ C(n).
n
t≤s≤T
The dominated convergence theorem implies
−
v(t, x) = E e
R T ∧τn
t
ρ(u,Xut,x )du
Stochastic Calculus 2 - Annie Millet
v(T ∧
τn , XTt,x∧τn )
=
2
X
Tni ,
i=1
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Diffusions and PDEs
h
i
R
− tτn ρ(s,Xst,x ) ds
t,x
1{τn ≤T } ,
= E v(τn , Xτn ) e
i
h
RT
t,x
Tn2 (t, x) = E v(T, XTt,x ) e− 0 ρ(s,Xs ) ds 1{τn >T } .
Tn1 (t, x)
There exist K > 0 such the term |Tn1 (t, x)| is upper estimated by CnK P (τn ≤ T ). Since for
any p ∈ [1, +∞[, E(supt≤s≤T |Xst,x |p ) ≤ Cp (1 + |x|p ), we deduce that for any p ∈ [1, +∞[,
P (τn ≤ T ) ≤ Cp n−p E sup |Xst,x |p ≤ Cp n−p .
t≤s≤T
Let p > K ; we deduce that Tn1 (t, x) → 0 as n → +∞. The dominated convergence theorem
implies that as n → +∞,
i
h
h
i
RT
RT
t,x
t,x
Tn2 (t, x) → E v(T, XTt,x ) e− 0 ρ(s,Xs ) ds = E f (XTt,x ) e− 0 ρ(s,Xs ) ds ) = v(t, x) ,
which concludes the proof of (2.33).
In order to prove (2.34), it suffices to notice that the martingale property of Mst,x , t ≤
s < T written for t = 0 yields for any n ≥ 1 and s ∈ [0, T [,
R s∧τn
Rt
e− 0 ρ(u,Xu )du v(t, Xt ) = E e− 0 ρ(u,Xu )du v(s ∧ τn , Xs∧τn Ft .
Let n → ∞ and s → T ; an argument similar to the previous one proves that
RT
RT
Rt
Rt
e− 0 ρ(u,Xu )du v(t, Xt ) = E e− 0 ρ(u,Xu )du f (XT )Ft = e− 0 ρ(u,Xu )du E e− t ρ(u,Xu )du f (XT )Ft ,
which yields (2.34)
2
In exercise 2.6 we will prove that the Cauchy problem (2.32) has
"
#
Z T
RT
Rs
t,x
t,x
v(t, x) = E f (XTt,x )e− t ρ(s,Xs ) ds +
g(s, Xst,x ) e− t ρ(u,Xu ) du ds
(2.35)
t
is a solution to the Cauchy problem.
The Feynman Kac Theorem immediately implies the following result. Let r > 0 and
f : Rd → R be a continuous function with polynomial growth. Then if v is continuous on
[0, T ] × Rd and v ∈ K1,2 ([0, T [×Rd ) is a solution the Cauchy problem
∂v
(t, x) + At v(t, x) = r v(t, x) for (t, x) ∈ [0, T [×Rd ,
∂t
v(T, x) = f (x) for x ∈ Rd ,
then v(t, x) = E e−r(T −t) f (XTt,x ) .
The relationship between diffusion process and PDEs can be seen in two ways.
One one hand, for « small dimension » one can use numerical methods for PDEs (finite
differences or finite elements) in order solve the PDE numerically (if it is known to have
a unique solution) and one can deduce from this an information on the expected value of
some function of the diffusion.
However, the numerical methods to solve PDEs are known to be efficient in small dimension, but are very difficult to implement in large dimension. On the other hand, the
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Stochastic differential equations
Monte-Carlo or quasi Monte Carlo methods allow to approximate numerically the expectation of a random variable or of a process in finitely many instants t and states x. This
is dimensionless and can be used to get information on the solution v(t, x) to the PDE
(provided that it is unique) by means of its probabilistic representation. Using an Euler
scheme, with mesh T /n, one can approximate the trajectory of the diffusion by that of a
process which is very easy and fast to simulate. The « strong » speed of convergence, which
gives an √
upper estimate of the uniform norm of the difference of both trajectories, is of
order 1/ n. If one is only interested in the expectation of a function of the diffusion at
some fixed time T , the « weak » speed of convergence of this scheme is of order 1/n. The
expected value of the function of the process at time T is approximated by an average of
independent realizations of the diffusion, using the strong law of large numbers. Then the
central limit theorem proves that when using N realizations, the speed of convergence of
the average to the expectation is of order √1N . The simulation of a diffusion by an Euler
scheme is very simple and very fast, but the drawback is rather the necessary number of
simulations which is necessary in order to obtain a « decent » approximation. Thus this a
« slow », but « dimension insensitive » method.
2.6 Examples in finance
We describe two basic examples of SDEs used in finance. More examples will be described
in a more systematic way in the last chapter.
2.6.1 The Sturm-Liouville - Occupation time
We have seen a link between diffusion processes and sont PDEs. Similar techniques can
also be used in order to connect some second order ODEs with stochastic processes. Let us
study the following so-called Sturm-Liouville problem :
f ′′ (x)
+ g(x)
(2.36)
2
where α > 0, k : R → R+ is a continuous function and g : R → R is a continuous function
such that :
Z
√
|g(x + y)|e−|y| 2α dy < +∞
(α + k(x))f (x) =
R
for all x ∈ R. Then if B denotes a standard Brownian motion starting from 0 and if
f ∈ K2 , then for any t > 0 the relation between the infinitesimal generator and a martingale
described in Theorem 2.17 with ρ(t, y) = α + k(x + y) ≥ α shows that the process
Z t R
R
− 0t α+k(x+Bs )ds
− 0s α+k(x+Bu )du 1 ′′
f (x+Bt ) e
−f (x)− e
f (x + Bs ) − α + k(x + Bs ) f (s, Bs ) ds
2
0
is an (FtB )-martingale. Therefore, if f ∈ C2 ∩ K1 denotes a bounded solution to (2.36), one
has
Z t R
h
i
R
− 0t α+k(x+Bs )ds
− 0s α+k(x+Bu )du
E f (x + Bt ) e
− f (x) = −E
e
g(x + Bs )ds .
0
As t → +∞, αt +
Rt
k(x + Bs ) ds → +∞ and if f is bounded, we deduce :
Z ∞
Z t
f (x) = E
g(x + Bt ) exp −αt −
k(x + Bs )ds dt .
0
0
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45
Examples in finance
Thus this function is the unique bounded solution of class C 2 to (2.36)(2.36) which belongs
to K1 .
This result gives in particular the Laplace transform of the random variable
Z t
g(Bt ) exp −
k(Bs )ds
0
for any function
inverse Laplace transform, we can deduce the distribution of
R g, and using
t
the pair Bt , 0 k(Bs )ds .
This pair plays an important role in finance. Indeed, it allows « pricing » some exotic
options with occupation time. The Feynman-Kac formula allows to compute the density of
the random variable
Z
t
A+
t =
1[0,∞[ (Bs ) ds
0
which denotes the occupation time of the interval [0, +∞[ by the Brownian motion. Indeed,
let k(x) = β1[0,+∞[(x) and g(x) = 1. Approximating k by a sequence of continuous functions,
we deduce that for α, β > 0 the function
Z ∞
Z t
f (x) = E
exp −αt − β
1[0,∞) (x + Bs )ds dt
0
is a solution to the ODE
(
0
αf (x) = 1 − βf (x) +
′′
αf (x) = 1 + f 2(x)
f ′′ (x)
2
if x > 0,
if x < 0.
The unique bounded and continuous solution to this ODE is
(
√
1
Ae−x 2(α+β) + α+β
if x > 0,
√
f (x) =
if x < 0.
Bex 2α + α1
Thus, imposing that both f and f ′ are continuous at 0, we deduce
A = p
1
1
−
α(α + β) (α + β)
which yields for x = 0
Z
Recall that Γ( 12 ) =
Z
∞
0
2009-2010
−αt
e
√
Z
0
i
h
+
1
e−αt E e−βAt dt = f (0) = p
.
α(α + β)
π. Using the Fubini theorem and a change of variable, we obtain
t
0
∞
1
1
and B = p
− ,
α(α + β) α
−βu
du √e
π u(t−u)
Z
+∞
e−(α+β)u
1
√
du √
u
π
0
1
1
√ , ∀α > 0.
=√
α+β α
1
dt = √
π
Z
u
+∞
e−α(t−u)
√
dt
t−u
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Stochastic differential equations
Since the distribution
characterized by its Laplace transform, we deduce that for any
R t is
−βA+
−βu √ 1
t
β > 0, E[e
]= 0e
du, and hence that the density of A+
t is the function h
π
u(t−u)
defined by
1
1]0,t[ (u).
h(u) = p
π u(t − u)
The repartition function of this distribution is defined for any θ ∈]0, t[ by :
P A+
t ≤ θ
=
Z
0
θ
π
p
du
u(t − u)
=
Z
0
θ/t
2
p
= Arcsin
π
π s(1 − s)
ds
r
θ
.
t
1 1
The distribution of A+
t (which coincides with that of tZ where Z has a β( 2 , 2 ) distribution)
is called the Arcsine distribution. Finally, note that for any θ ∈]0, t[,
P A+
= P tA+
t ≤ θ
1 ≤ θ ,
+
which proves that the random variables A+
t and tA1 have the same distribution. This result
can also be derived directly from the « scaling » property of Brownian motion.
2.6.2 Introduction to the Black & Sholes formula
We give a short introduction of this fundamental example which we will study in more
details in the last chapter. Let us consider a financial market with a non risky asset with
constant interest rate r, such that S00 = 1. Thus the price at time t > 0 is St0 = ert (or in
other words, dSt0 = rSt0 dt). The financial market also contains a risky asset with price St at
time t is
St = S0 exp σBt + (b − σ 2 /2)t .
(2.37)
This is the geometric Brownian motion already defined in Example 2.7, solution to the linear
SDE
dSt = σ St dBt + b St dt ,
where S0 > 0, B is a Brownian motion and b, σ ∈ R. Note that if S0 > 0, this process keeps
strictly positive values at any time t > 0, E(St ) = S0 ebt . The expected value of this process
St is that on a non risky asset with constant rate b. Furthermore,
(2σ)2
2
2
2 2bt+σ2 t
2σBt − 2 t
E(St ) = S0 e
E e
= S02 e2bt+σ t ,
2
which yields V ar(St ) = S02 e2bt eσ t − 1 . The coefficient σ is called the volatility ; it measures the sensitivity to the random perturbation, that is the risk one takes investing on this
asset. Indeed, it measures the difference between the price St of the asset and its expected
value. Using the explicit formulation of St , it is easy to see that for any fixed δ > 0 which re
S(j+1)δ
present a time interval (for example one day, one hour, ...), the sequence ln Sjδ
,j ≥ 0
2
is an i.i.d. sequence of Gaussian random variables N (b − σ2 )δ ,, σ 2 δ . Therefore, it is easy
to check if the model is valid
with actual data. A Taylor expansion of ln(1 +u)
by comparing
S(j+1)δ
S(j+1)δ −Sjδ
allows to approximate ln Sjδ , by
to test whether the successive rates are
Sjδ
indeed i.i.d. Gaussian random variables. Doing so, one can notice that the model is not
adequate.
Stochastic Calculus 2 - Annie Millet
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47
Examples in finance
If someone dislikes taking risks, he will only agree to invest money on the risky asset
when its expected rate b is larger that that r of the non risky asset and the difference b − r
is called the « risk prime ». The larger the volatility σ, the more risky the asset is and in
that case b has to be large to make the investor prefer the risky asset St to the non risky
one St0 .
Let us fix finite horizon T > 0. One wants to find the price of a financial asset which
will pay h(ST ) at time T . The case of a European call of strike K and maturity T is given
by h(x) = (x − K)+ .
One proceeds using replication (hedging) : one creates a portfolio containing αt shares
of the non risky asset St0 (that is, the amount of money invested in this asset at time t is
αt ert ) and of ∆t shares of the risky asset St .
One assumes that the market is frictionless, that is there is no transaction cost (for selling
of buying stocks) that one may have negative investment (that means that both coefficients
αt and ∆t may be negative reals), that the shares are infinitely divisible (αt and ∆t are
real-valued) and that trading is taking place continuously (that means that αt and ∆t can
change at any time t).
One wants to find a self-financing portfolio, (that is money is put in only at the initial
time 0), which does not pay dividend (that means that no money is withdrawn before the
terminal time T ) and with terminal value h(ST ). The value of this portfolio at time t is
Vt = αt St0 + ∆t St .
The self-financing property can be described by the following evolution of the value of the
portfolio
dVt = αt dSt0 + ∆t dSt ,
that is
dVt = αt rSt0 dt + ∆t [bSt dt + σSt dBt ] = σ∆t St dBt + rVt + ∆t St (b − r) dt.
The initial value of the portfolio will be called the value. We assume that the value Vt of
the portfolio at time t is a deterministic function of time and of the price of the risky asset,
that is Vt = V (t, St ). Using Itô’s formula we deduce
∂V
∂V
σ 2 St2 ∂ 2 V
∂V
dVt =
(t, St ) + b St
(t, St ) +
(t, St ) dt + σSt
(t, St ) dBt .
∂t
∂x
2 ∂x2
∂x
Thus an identification of martingale parts (using the fact that St > 0 for any t) yields
σ∆t St = σSt
∂V
∂V
(t, St ) that is ∆t =
(t, St ) .
∂x
∂x
Then identifying the finite variation parts we deduce
rSt
∂V
σ 2 St2 ∂ 2 V
∂V
(t, St ) +
(t, St ) +
(t, St ) − rV (t, St ) = 0 ,
∂x
∂t
2 ∂x2
with terminal condition V (T, ST ) = h(ST ). As the random variable St may take all positive
values, we deduce that V satisfies on [0, +∞[×]0, +∞[ the PDE :
rx
2009-2010
∂V
σ 2 x2 ∂ 2 V
∂V
(t, x) +
(t, x) +
(t, x) − rV (t, x) = 0 ,
∂x
∂t
2 ∂x2
(2.38)
Stochastic Calculus 2 - Annie Millet
48
2
Stochastic differential equations
with terminal condition V (T, x) = h(x). Note that the drift coefficient of the risky asset has
disappeared.
We will solve this PDE in the particular case of a European call, that is when the terminal
condition is h(x) = (x − K)+ , with σ > 0. The function obviously belongs to h ∈ K0 . Let
C denote the value of the portfolio in this particular case. The detailed computations are
leaved to the reader as an exercise (cf. Exercise 2.5). We will see in the next chapter a
simpler and less technical way to recover them.
The previous computations prove that C satisfies equation (2.38)
∂C
∂C
σ 2 x2 ∂ 2 C
(t, x) + rx
(t, x) +
(t, x) = rC(t, x) ,
∂t
∂x
2 ∂x2
with C(T, x) = (x − K)+ . Using the Feynman-Kac formula (2.33) with the function f (x) =
(x − K)+ and the discounting factor ρ(s, x) = r we deduce that the value of the European
call at time t is
i
h
r(t−T ) et,x
+
(2.39)
C(t, x) = E e
(ST − K) ,
where
SeTt,x = xeσ(BT −Bt )+
2
r− σ2
(T −t)
solves the SDE dS̃st,x = σ S̃st,x dWs + bS̃st,x ds for s ≥ t and S̃tt,x = x.
The law of the random variable BT − Bt is Gaussian N (0, T − t), and we can compute
C(t, x) explicitly. For a ∈ R, let
Z a
x2
1
e− 2 dx
F (a) = √
2π −∞
denote the repartition function of a standard Gaussian random variable N (0, 1). Then the
solution to (2.39) is given by the following Black & Sholes formula
C(t, x) = xF (d1 ) − Ke−r(T −t) F (d2),
with the notations
1
d1 = √
σ T −t
Clearly, we have
1 2
r(T −t)
ln xe
/K + σ (T − t)
2
√
et d2 = d1 − σ T − t.
(2.40)
∂C
(t, x) = F (d1).
∂x
Furthermore,
∂C
(t, St )
∂x
which represents the number of shares of the risky asset which is needed to replicate the
option is called the Delta for the option. It also represents the sensibility of the option
with respect to the risky asset’s price. The pair (C(t, St ) − ∆t St , ∆t ) describes the hedging
portfolio.
∆t =
Furthermore, the Feynman-Kac Theorem 2.24 applied with ρ(s, x) = r and f (x) =
(x − K)+ proves when using (2.34) that if Ses = Ss0,x is a solution to dSet = σ Set dBt + r Set dt
with Se0 = x, then
h
i
−r(T −t)
+
e
e
C(t, St ) = e
E (ST − K) |Ft
(2.41)
Stochastic Calculus 2 - Annie Millet
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Exercises
Formula (2.39) makes it possible to recover the Delta differently. Indeed, for t = 0, one has
C(0, x) = E e−rT (xMT erT − K)+ ) ,
2
σ
e
with the notation Set = xMt ert and when Mt = SS0t = x eσBt − 2 t for t ≥ 0 is a martingale.
t
Differentiating with respect to x under the expected value, we get again
∂C
(0, x) = E MT 1{xMT erT ≥K} = F (d1 ),
∂x
where d1 is the above constant computed for t = 0.
2.7 Exercises
Exercise 2.1 This exercise proposes two extensions of the Gronwall Lemma.
(i) Let f : [0, T ] → R be a continuous function. Suppose that there exists a constant
b > 0 and a non negative function a : [0, T ] → R such that for any t ∈ [0, T ],
Z t
0 ≤ f (t) ≤ a(t) + b
f (s)ds.
(2.42)
0
Rt
Then f (t) ≤ a(t) + b 0 a(s)eb(t−s) ds for all t ∈ [0, T ].
(ii) Let f : [0, T ] → R be a continuous Borel function. Suppose that there exists a
constant
function b : [0, T ] → R such that 0 ≤ f (t) ≤
R t a > 0 and an integrable non negative
Rt
a + 0 b(s)f (s)ds. Prove that for H(t) = 0 b(s)ds, one has f (t) ≤ aeH(t) for any t ∈ [0, T ].
Exercise 2.2 The Ornstein-Uhlenbeck process Let X0 , σ and b denote real numbers,
(Bt ) be a one-dimensional standard Brownian motion, X denote the solution to the Langevin
equation
Z
t
Xt = X0 + σBt − b
Xs ds.
0
1. Prove that if X 1 et X 2 are solutions to the Langevin equation, X 1 = X 2 .
Rt
2. Let Yt = e−bt X0 + σ 0 ebs dBs . Prove that
−bt
Yt = e
Z t
bs
X0 − bσ
e Bs ds + σBt .
0
Using Fubini’s theorem, deduce that
Z t
Z t
−bt
b
Ys ds = X0 1 − e
+ bσ
e−b(t−u) Bu du = X0 − Yt + σBt .
0
0
Deduce that the unique solution to the Langevin equation is
Z t
bs
−bt
X0 + σ
e dBs .
Xt = e
0
3. Prove that for all t, the distribution of Xt is Gaussian N
all s < t, compute Cov(Xs , Xt ).
2009-2010
2
X0 e−bt , σ2b (1
−2bt
−e
) . For
Stochastic Calculus 2 - Annie Millet
50
2
Stochastic differential equations
Exercise 2.3 The Vasicek process One generalizes the previous process by adding a
constant in the drift term.
dYt = σdBt + a(b − Yt )dt.
This SDE has been studied to model the evolution of interest rates on a financial market
and is called « mean-reversing » because if a and b are positive, the process Yt goes to b (in
a way to be made precise).
1. Prove that Xt = Yt − b is a solution to the Langevin equation.
2. Suppose that Y0 is constant. Deduce an explicit expression of Yt , find the distribution
of E(Yt ) and Cov(Ys , Yt ) for 0 ≤ s ≤ t.
3. Suppose that Y0 has a Gaussian N (m0 , σ02 ) distribution and is independent of (Bt ).
Answer the previous question and compute the distribution of the vector (Ys , Yt) for
0 ≤ s < t, the expected value of Yt and the covariance of Ys and Yt . If Y0 > 0, does
the process Yt keep non negative
values
R ?
T
Let 0 ≤ t ≤ T , and compute E exp t Yu du FtB .
Exercise 2.4 Let (Bt ) be a standard r-dimensional (Ft )-Brownian motion, (σk (s), s ≥ 0)
et (b(s), s ≥ 0) be real progressively measurable processes defined for k = 1, · · · , r, and M
be a real constant such that sup0≤t≤T (kσ(s)k2 + |b(s)|) ≤ M a.s. For any X0 ∈ R one wants
to find an explicit solution to the linear SDE (with non constant coefficients)
Z
Xt = X 0 +
t
σ(s)Xs dBs +
0
Z
t
b(s)Xs ds.
(2.43)
0
1. Suppose that there exists a real constant M telle que sup (kσ(s)k2 + |b(s)|) ≤ M
0≤t≤T
a.s. Along the line of the proof of Theorem 2.6 prove that (2.43) has a unique solution
such that sup0≤t≤T E(|Xt |2 ) < +∞. Under which conditions on the coefficients does
the process satisfy the Markov property ?
2. Let the assumptions of the previous question be satisfied. Let
Z t
Z t
1
2
Yt = exp −
σ(s)dBs +
−b(s) + kσ(s)k ds .
2
0
0
Compute dYt and then d(Xt Yt ) and deduce that
Xt = X0 exp
Z
t
σ(s)dBs +
0
Z t
0
1
2
b(s) − kσ(s)k ds .
2
(2.44)
Exercise 2.5 The aim of this exercise is to give a first approach to prove equation (2.40)
in the celebrated Black & Sholes model.
1. Show that the solution to the Black & Sholes equation (2.38) is
∂C
∂C
σ 2 x2 ∂ 2 C
(t, x) + rx
(t, x) +
(t, x) = rC(t, x)
∂t
∂x
2 ∂x2
with C(T, x) = (x − K)+ .
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Exercises
2. Prove that the value of the European call at time t is given by (2.39)
h
i
C(t, x) = E er(t−T ) (SeTt,x − K)+ ,
where SeTt,x = xeσ(BT −Bt )+
2
r− σ2
(T −t)
√
.
2
3. Prove that if G ∼ N (0, 1), α = σ T − t > 0 and β = r − σ2 (T − t),
h
i
E e−r(T −t) (SeTt,x − K)+ = xe−r(T −t) E eα G+β 1{α G+β≥ln(K/x)}
−Ke−r(T −t) P (α G + β ≥ ln(K/x)) .
Rx
2
4. Let F (x) = √12π −∞ e−u /2 du denote the repartition function of a standard Gaussian
N (0, 1) random variable G. Prove that for all α > 0, β ∈ R, K > 0 et x > 0,
ln(x/K) + β
P α G + β ≥ ln(K/x) = F
α
and
αG+β
E e
2
β+ α2
1{αG+β>ln(K/x)} = e
5. Prove the formula (2.40).
ln(x/K) + β
F α+
α
Exercise 2.6 The aim of this exercise is to generalize the Feynman-Kac formula proved in
Theorem 2.24 by proving (2.35).
Let σ and b be coefficients which satisfy the global Lipschitz and growth conditions
(2.3) and (2.4), f, g, ρ be functions which satisfy the conditions of Definition 2.23. Suppose
that f ∈ K0 (Rd ) and that sup0≤t≤T |g(t, x)| have polynomial growth. For any t ∈ [0, T [ and
x ∈ Rd , let (Xst,x , s ∈ [t, T ]) denote the process solution to (2.31) and (Ft = FtB ) the natural
filtration of the Brownian motion B. Then a solution v to the Cauchy problem (2.32)
∂v
(t, x) + At v(t, x) − ρ(t, x) v(t, x) + g(t, x) = 0 pour (t, x) ∈ [0, T [×Rd ,
∂t
v(T, x) = f (x) pour x ∈ Rd .
has the following stochastic representation :
"
v(t, x) = E f (XTt,x )e−
RT
t
ρ(s,Xst,x ) ds
+
Z
T
t
g(s, Xst,x) e−
Rs
ρ(u,Xut,x ) du
t
#
ds .
The following five exercises study in a systematic way the solution to the SDE
dXt = (αXt + a)dBt + (βXt + b)dt , X0 = x.
Exercise 2.7 Preliminary
Let a, α, b, β be real non negative constants and x ∈ R. Consider the SDE
dXt = (αXt + a)dBt + (βXt + b)dt , X0 = x.
(2.45)
1. Prove that (2.45) has a unique strong solution.
2. Let m(t) = E(Xt ) and M(t) = E(Xt2 ).
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2
Stochastic differential equations
(a) Prove that m(t) is a solution to the ODE
y ′ − βy = b , y(0) = x.
(2.46)
Deduce that m(t).
(b) Write Xt2 as an Itô process. Prove that the stochastic integral is a square integrable martingale.
(c) Deduce that if m(t) is a solution to (2.46), M(t) is a solution to the following
ODE
y ′ − (2β + α2 )y = 2(b + aα)m + a2 , y(0) = x2 .
Deduce the value of M(t).
Exercise 2.8 First particular case a = b = 0
Let (Yt ) be the solution to (2.45) with a = b = 0 et x = 1.
“
”
2
αBt + β− α2 t
1. Show that Yt = e
.
2. Prove that β ≥ 0, (Yt ) is a (Ft )-submartingale. Under which condition on β is the
process (Yt ) a martingale ?
3. Let (Zt ) denote the Itô process defined by
Z t
Z t
−1
Zt = x + a
Ys dBs + (b − aα)
Ys−1 ds.
0
0
Prove that this process is well-defined and compute hY, Zit . Deduce that the solution
(Xt ) to (2.45) can be written as Xt = Yt Zt .
Exercise 2.9 Second particular case α = b = 0
Let (Xt ) be the solution to the SDE
dXt = adBt + βXt dt , X0 = x.
(2.47)
1. Prove that the unique solution to (2.47) is
Z t
βt
−βs
x+a
Xt = e
e dBs .
0
(One may introduce Yt = e−βt Xt and solve the SDE satisfied by Y .)
2. Prove that for any instants t1 < t2 < · · · < td , the vector ξ = (Xt1 , · · · Xtd ) is Gaussian.
Compute E(ξ) and the covariance matrix of ξ.
3. Compute E(Xt |Fs ) and V ar(Xt |Fs ) for all 0 < s < t.
2
4. Let (Xt ) denote the solution to (2.47) and let φ be a function
of
class C . Write the
Rx
2
Itô formula for Zt = φ(Xt ). Deduce that if φ(x) = 0 exp −β ay2 dy, one has
Zt = φ(x) + a
Z
0
t
X2
exp −β 2s
a
dBs .
Let β ≥ 0 ; is the process (Zt ) a square integrable martingale ?
Stochastic Calculus 2 - Annie Millet
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53
Exercises
λXt2
5. Fix λ ∈ R. Compute E e
.
Exercise 2.10 Third particular case b = β = 0
Let (Xt ) denote the solution to the SDE
a
dXt = (αXt + a)dBt , X0 = x 6= − .
α
(2.48)
Let h be the function defined by
1 a + αy a
h(y) = ln for y 6= − .
α
a + αx
α
1. Set Yt = h(Xt ). What is the SDE satisfied by Yt ?
2. Deduce that the solution to (2.48) is
α2
a
a
exp − t + αBt − .
Xt = x +
α
2
α
Exercise 2.11 Fourth particular case b = 1 and a = 0
Let (Xt ) be the solution to (2.45) when b = 1 and a = 0. Set Yt = e−βt Xt . What is the SDE
satisfied by Yt ? Compute E(Xt ) and V ar(Xt ) when α2 6∈ {−β, −2β}.
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3
The Girsanov Theorem
3 The Girsanov Theorem
Together with the Itô formula, the Girsanov theorem is a fundamental tool in stochastic analysis. It describes a way to change the original probability into another absolutely
continuous one and how the distribution of some processes is modified in the mean time.
More precisely, let (Xt , t ≥ 0) be a process defined on the probability space endowed with
a filtration (Ω, F , (Ft ), P ), and suppose that the distribution of Xt under P is « complicated ». One « perturbs » the measure P by means of an (Ft ) exponential martingale density.
This yields a new probability Q under which the distribution of the process X is different
and simpler to deal with. Thus, this distribution is studied under Q and then using the
exponential martingale one comes back to P . This method has many applications, in the
framework of finance or to study properties of the Brownian motion.
Convention In this chapter we will have to use simultaneously two
R probabilities on a measurable space (Ω, F ). If P denotes such a probability, let EP (X) = XdP denote the expected
value of a non negative or P-integrable random variable X, and similarly let EP (X | G) denote the conditional expectation of X given the sub σ-algebra G of F under the probability
P.
3.1 Changing probability
Let (Ω, F , P) be a probability space and let L be a non negative F -measurable random
variable such that EP (L) = 1, that is the expected value of X under P is equal to one.
Let Q denote the probability defined on F with density L with respect to P as follows : set
Q(A) = EP (L1A ) for every A ∈ F ; one sets dQ = L dP. A random variable X is Q integrable
if and only if LX is P-integrable. If X is non negative or Q-integrable, EQ (X) = EP (LX).
Furthermore, Q is absolutely continuous with respect to P, that is for any A ∈ F , P(A) = 0
implies Q(A) = 0. Furthermore, if L > 0 P a.s., then the probability P has for density L1
with respect to Q, P is absolutely continuous with respect to Q, which means that both
probabilities P and Q are equivalent, that is they have the same null sets.
Lemma 3.1 Let (Ω, F , (Ft), P) be a probability space endowed with a filtration, T > 0. Let
LT denote a non negative FT -measurable random variable such that EP (LT ) = 1. For any
t ∈ [0, T ], set Lt = EP (LT | Ft ). (i) For t < T , let dQ|Ft (resp. dP|Ft ) denote the restriction
of Q (resp. P) to Ft . Then
dQ|Ft = Lt dP|Ft .
(ii) Let (Mt , t ≥ 0) denote a stochastic process. It is a (Ft , t ∈ [0, T ])-martingale (resp. a
(Ft , t ∈ [0, T ]) local martingale) under the probability Q if and only if the process (Lt Mt , t ∈
[0, T ]) is a (Ft , t ∈ [0, T ])-martingale (resp. a (Ft , t ∈ [0, T ]) local martingale) under the
probability P.
(iii) If LT > 0 P a.s., the probabilities P and Q are equivalent, for any t ∈ [0, T ], Lt > 0
P (or Q) almost surely, ( L1t , 0 ≤ t ≤ T ) is a (Ft , 0 ≤ t ≤ T ) martingale under Q and if Z
is Ft measurable and either non negative or Q-integrable, then for any s ∈ [0, t],
EQ (Z | Fs) =
EP (ZLt | Fs )
EP (ZLt | Fs )
=
.
Ls
EP (Lt | Fs)
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The Cameron Martin formula
Proof. The filtration (Ft , t ∈ [0, T ]) is fixed and all the martingales that we will consider
(under either probability P or Q) will always be martingales for this filtration without telling
so.
(i) For t < T and any A ∈ Ft one has
Q(A) = EP (LT 1A ) = EP (EP [LT | Ft ] 1A ) = EP (Lt 1A ) .
(ii) Suppose that (Mt , t ∈ [0, T ]) is a Q-martingale. For any s ≤ t ≤ T and any A ∈ Fs
one has
EP (Lt Mt 1A ) = EQ (Mt 1A ) = EQ (Ms 1A ) = EP (Ls Ms 1A )
and this yields that (Lt Mt , t ∈ [0, T ] is a P-martingale. The proof of the converse implication
is similar. Let M be a local martingale under Q, τn → ∞ denote an increasing sequence
of stopping times such that Mτn ∧t is a (Ft )-martingale under Q. Then (Mτn ∧t Lτn ∧t ) is a
(Ft )-martingale under P, which completes the proof.
(iii) Since LT > 0, the probability P is absolutely continuous with respect to Q and
its density on FT is L1T . For any t ∈ [0, T ], the random variable L1t : Ω → [0, +∞] is non
negative and EQ (1/Lt ) = EP (Lt /Lt ) = 1. Therefore, L1t < +∞ Q a.s., that means Lt > 0 Q
(and P) a.s.
Let Z ∈ L1 (Q) be a Q-integrable, Ft -measurable random variable. Then, for s ≤ t,
−1
Ls EP (ZLt |Fs ) is Fs -measurable and for any A ∈ Fs ,
EQ (Z1A ) = EP (Z1A Lt ) = EP (1A EP (ZLt | Fs)) = EQ (1A L−1
s EP (ZLt | Fs )),
which implies that EQ (Z|Fs ) = L−1
s EP (ZLt |Fs ) Q a.s. (or P a.s.) The above argument is
similar if Z ≥ 0 is Ft measurable.
Let us prove that for t ∈ [0, T ], L−1
= EQ (L−1
t
T | Ft ) Q a.s. Let Z be a bounded Ft −1
measurable random variable. Then alors Z Lt is Ft measurable and
−1
−1
EQ (ZL−1
t ) = EP (ZLt Lt ) = EP (Z) = EQ (ZLT ).
−1
Since L−1
is Ft -measurable, L−1
t
t = EQ (LT |Ft ) Q a.s.
2
3.2 The Cameron Martin formula
Let us at first present the main idea in finite dimension. Let X1 , . . . , Xn be independent standard Gaussian variables on the same probability space (Ω, F , P). For every
(µ1 , . . . , µn ) ∈ Rn one has
"
!#
!
n
n
n
X
Y
1X 2
EP exp
µ i Xi
=
EP [exp (µi Xi )] = exp
µi ,
2
i=1
i=1
i=1
which implies that
"
n X
µ2i
EP (L) = 1 where L = exp
µ i Xi −
2
i=1
#
.
Thus, one may define a new probability Q on (Ω, F ) as follows : dQ = L dP, that is
Z
Q (A) = EP (L1A ) =
L(ω)dP(ω)
A
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The Girsanov Theorem
for all A ∈ F . Lemma 3.1 shows that Q is a probability equivalent with P on (Ω, F ), that
is P (A) = 0 if and only if Q (A) = 0 for any A ∈ F . the random variable L denotes the
density of Q with respect to P.
One wants to find the distribution of the random vector (X1 , . . . , Xn ) under this new
probability Q. Thus, for every Borel subset A ∈ Rn , one has
„
«
Z
Pn
µ2
i
µ
X
−
i
i
i=1
2
Q (X1 , · · · , Xn ) ∈ A =
1{(X1 ,··· ,Xn )∈A} e
dP
= (2π)
−n/2
= (2π)−n/2
Z
ZA
A
Pn
e
i=1
1
e− 2
„
Pn
µi xi −
µ2
i
2
«
e−
2
i=1 (xi −µi )
x2
i
i=1 2
Pn
dx1 · · · dxn
dx1 · · · dxn .
Let µ = (µ1 , . . . , µn ) : under Q, (X1 , . . . , Xn ) is a Gaussian vector N (µ, Id), that is :
under Q, (X1 , · · · , Xn ) − (µ1 , µ2 , · · · , µn ) is a Gaussian vector N (0, Id), i.e., the vectors
(X1 , . . . , Xn ) under P and (X1 , · · · , Xn ) − (µ1 , µ2 , · · · , µn ) under Q have the same distribution.
The Cameron-Martin formula obeys the same principle of probability change, except that
this time it occurs on the space of continuous functions, that is on an infinite dimensional
space. Let (Bt , t ≥ 0) be a standard Brownian motion on the probability space (Ω, F , P),
and let (FtB , t ≥ 0) denote its natural (complete) filtration. For any m ∈ R, the process
m2 t
m
t 7→ Lt = exp mBt −
is a (FtB )positive martingale.
2
This martingale Lm is clearly similar to the previous random variable L. Fix a finite horizon
B
m
T > 0 and let Qm
be defined by Qm
T be a new probability on Ω, FT
T (A) = EP (LT 1A ) .
m
The Cameron-Martin gives the distribution of B under QT .
Theorem 3.2 (The Cameron-Martin formula) Let B be a standard Brownian motion and
let m ∈ R. With the above notations, the process B̃ defined by (B̃t = Bt − mt, 0 ≤ t ≤ T )
is a (FtB , 0 ≤ t ≤ T ) standard Brownian motion under the probability Qm
T.
B̃
B
Proof. Notice at
first that the
σ-algebras Ft and Ft coincide for all t ≤ T . Fix λ ∈ R and
let Ztλ = exp λB̃t − λ2 t/2 . For any s ≤ t ≤ T and any A ∈ FsB , one has :
m2 t
λ2 t
λ
λ m
1A
+ mBt −
EQ Zt 1A = EP Zt Lt 1A = EP exp λB̃t −
2
2
m2 t
λ2 t
1A
+ mBt −
= EP exp λBt − λmt −
2
2
(λ + m)2 t
= EP exp (λ + m)Bt −
1A
2
h
i
2t
If B is a P Brownian motion, the process exp (λ + m)Bt − (λ+m)
,
t
∈
[0,
T
]
is a P2
(FtB )-martingale. Hence, for every A ∈ FsB ,
(λ + m)2 s
λ
1A
EQ Zt 1A = EP exp (λ + m)Bs −
2
= EP Zsλ Lm
= EQ .Zsλ 1A
s 1A
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The Girsanov Theorem
This implies EQ Ztλ 1A = EQ Zsλ 1A , and hence EQ Ztλ | FsB = Zsλ Q a.s. for any
s ≤ t ≤ T . For any λ ∈ R, the process Z λ is a (FtB )-martingale under Q and the Paul
Lévy Theorem 1.33 (about the characterization of Brownian motion) yields that B̃ est un
(FtB , 0 ≤ t ≤ T ) is a Brownian motion under Q (also called Q Brownian motion).
2
3.3 The Girsanov Theorem
The aim of this section is to extend the Cameron Martin theorem, translating the Brownian trajectory Bt by means of the integral at time t of a non-constant process. This theorem
has many applications.
Theorem 3.3 Let (Bt , t ≥ 0) denote a standard d-dimensional Brownian motion on
(Ω, F , P) et (FtB , t ≥ 0) denote its natural (complete) filtration. Let T > 0, (θ(t), 0 ≤ t ≤ T )
be a d-dimensional process which belongs to H2T and for any t ∈ [0, T ], let
!
Z tX
Z t
n
1
Lt = exp
θi (s)dBsi −
kθ(s)k2 ds .
(3.1)
2
0 i=1
0
Then if EP (LT ) = 1, the process (Lt , 0 ≤ t ≤ T ) is a (FtB )-martingale, and
Q(dω) = LT (ω)P(dω) is a probability equivalent to Q
The process W = (Wti , 0 ≤ t ≤ T , i = 1, · · · , d) defined by
Wti
=
Bti
−
Z
t
θi (s)ds
0
is a (FtB ) Brownian motion under Q.
(3.2)
Proof. To highlight the ideas in the proof, we will restrict ourselves to the case d = 1. Let
θ = (θt ) ∈ HRT2 and (Lt , 0 R≤ t ≤ T ) denote the process defined by (3.1). Then Lt = eXt > 0
t
t
where Xt = 0 θs dBs − 21 0 θs2 ds is an Itô process. The Itô formula under P proves that
1
dLt = Lt dXt + Lt θt2 dt = Lt θt dBt .
2
Since θ ∈ H2T and the trajectories of Lt are a.s. continuous, they are a.s. bounded on the
interval [0, T ] ; thus the process θt Lt belongs to H2loc . This implies that under P, (Lt , t ∈
[0, T ]) is a non negative local martingale. Hence it is a super-martingale (see Exercise 1.4).
Therefore, for 0 ≤ t ≤ T , Lt ≥ EP (LT |FtB ). Furthermore, we have EP (LT ) = 1 = EP (L0 )
by hypothesis. Therefore, EP (Lt ) = 1 for every t ∈ [0, T ] and (Lt , 0 ≤ t ≤ T ) is a (FtB )martingale under P.
Using the characterization of Brownian motion proved in the Paul Lévy Theorem 1.33
(ii), since W0 = 0, in order to prove that W is a (FtB , 0 ≤ t ≤ T )-Brownian motion under
Q, it sufficed to prove that under Q the processes (Wt , 0 ≤ t ≤ T ) and (Wt2 − t, 0 ≤ t ≤ T )
are continuous (FtB ) local martingales. Using Lemma 3.1, this comes down to checking that
the processes (Lt Wt , 0 ≤ t ≤ T ) and (Lt (Wt2 − t), 0 ≤ t ≤ T ) are continuous (FtB ) local
martingales under P. Using Itô’s formula for a product and the identity
dWt = dBt − θt dt,
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The Girsanov Theorem
we deduce that under P,
d(Lt Wt ) = Lt dWt + Wt dLt + dhL, W it
= Lt dBt − Lt θt dt + Wt θt Lt dBt + Lt θt dt = (1 + θt Wt )Lt dBt .
This yields that the process (Lt Wt , 0 ≤ t ≤ T ) is a continuous (FtB , 0 ≤ t ≤ T ) local
martingale under P. Furthermore, the Itô implies that if Yt = Wt2 − t, under P,
1
dYt = 2Wt dWt + 2dhW, W it − dt = 2Wt dBt − 2Wt θt dt + dt − dt = 2Wt dBt − 2Wt θt dt.
2
Using again the Itô formula for a product, we deduce that under P
d(Lt Yt ) = Lt dYt + Yt dLt + dhL, Y it
= 2Lt Wt dBt − 2Lt Wt θt dBt + Yt Lt θt dBt + 2Wt Lt θt dt = Lt (2Wt + Yt θt )dBt .
2
This implies that the process (Wt − t)Lt , 0 ≤ t ≤ T is a continuous (Ft ) local martingale
under P.
2
Lemma 3.1 and Theorem 3.3 prove that P can be expressed in terms of Q as follows :
dP = L−1
integral with respect to B into a stochastic
T dQ. Let us change the stochastic
Rt
integral with respect to Wt = Bt − 0 θs ds. This yields
Z t
Z t
Z
Z
1 t 2
1 t 2
−1
Lt = exp −
θs dBs +
θ ds = exp −
θ ds ,
θs dWs −
2 0 s
2 0 s
0
0
where the last exponential of a stochastic integral with respect to the process W , which is
a Q Brownian motion, is clearly a (Ft , 0 ≤ t ≤ T )-martingale under Q.
used as follows. Let θ ∈ H2T ; for all t ∈ [0, T ] let Zt =
R t The Girsanov theorem is often
1
θ dBs and Lt = exp Zt − 2 hZ, Zit . Suppose that E(LT ) = 1 and let Q denote the
0 s
probabilityR whose restriction to Ft is Lt for all t ∈ [0, T ]. On the other hand let H ∈ H2loc
t
and Mt = 0 Hs dBs be a continuous (Ft ) local martingale under P, with bracket hM, Mit =
Rt
kHs k2 ds. Then the process Nt = Mt −hM, Zit is a continuous (Ft ) local martingale under
0
Q, with bracket hN, Nit = hM, Mit (under P or Q, since the brackets are the quadratic
variations and since the probabilities P are Q equivalent, so that the brackets are the same
under both probabilities).
Indeed, Lemma 3.1 proves that in order to show this property, it suffices to check that,
under P, the process Lt Nt = Lt (Mt − hM, Zit ) is a (Ft ) local martingale, which is a straightforward consequence of the Itô formula.
3.4 The Novikov condition and some generalizations
The following theorem will be very useful in the next chapter. It introduces a local
exponential martingale and gives sufficient condition for it to be a martingale. This is the
key ingredient to define a fundamental change of probability in finance.
Rt
Let θ ∈ H2loc (FtB ) be a cad-lag, (FtB )-adapted process, such that 0 θs2 ds < +∞ a.s. for
all t > 0 and Z0 is a constant. Using Itô’s formula, we prove that the unique solution to the
SDE
Z t
Zt = Z0 +
θs Zs dBs
(3.3)
0
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The Novikov condition and some generalizations
is
Zt = Z0 exp
Z
t
0
1
θs dBs −
2
Z
t
0
θs2 ds
.
The process Z is called the Doléans-Dade exponential of θ ⋆B. Using (3.3), we deduce that it
is a non-negative local martingale. However, its expected value need not be constant (equal
to 1) and therefore it need not be a martingale. The following criterion allows to deduce
that the Doléans-Dade exponential is a martingale. It is formulated in a multidimensional
setting. One can read its proof in [7], p. 200.
Theorem 3.4 (The Novikov condition) Suppose that B is a (Ft ) standard d-dimensional
Brownian motion, (θt ) is a Rd -valued process, (Ft )-adapted such that
Z T
1
2
kθs k ds
< ∞.
E exp
2 0
For any t ∈ [0, T ] let
Lt (θ) = exp
d Z
X
i=1
0
t
1
θsi dBsi −
2
Z
0
t
!
kθs k2 ds .
The process (Lt (θ), 0 ≤ t ≤ T ) is a (Ft , 0 ≤ t ≤ T ) uniformly continuous martingale.
When the Novikov condition is not satisfied, Lt (θ) is a non negative local martingale, thus
a super-martingale, and E(Lt (θ)) ≤ E(Ls (θ)) ≤ L0 (θ) for t ≥ s ≥ 0. Notice that even
if the family (Lt (θ)) is uniformly integrable, it need not be a martingale. However, if the
family of random variables {Lτ (θ), τ stopping times} is uniformly integrable, then (Zt (θ))
is a uniformly integrable martingale.
Proposition 3.5 Let B denote a standard d-dimensional Brownian motion, θ ∈ H2T be an
Rd -valued process such that for some instants 0 = t0 < t1 < tN = T and any n ∈ {1, · · · , N},
Z tn
1
2
EP
kθs k ds < +∞.
(3.4)
2 tn−1
Rt
For t ∈ [0, T ] let Lt = e
is an (Ft )-martingale.
0
θs dBs − 21
Rt
0
kθs k2 ds
. Then EP (LT ) = 1 and the process (Lt , t ∈ [0, T ])
Proof. For n ∈ {1, · · · , N}, set θ(n)t = θt 1[tn−1 ,tn [ (t). The Novikov condition holds for the
process θ(n) and Theorem 3.4 shows that if one lets
Z t
Z
1 t
2
Lt (θ(n)) = exp
θ(n)s dBs −
kθ(n)s k ds ,
2 0
0
then one has
EP Ltn (θ(n))|Ftn−1 = Ltn−1 (θ(n)) = 1.
We prove EP (Ltn ) = 1 by induction on n ≤ N. Indeed, for n ∈ {1, · · · , N}
EP (Ltn ) = EP Ltn−1 EP Ltn (θ(n))|Ftn−1 = EP (Ltn−1 ).
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The Girsanov Theorem
Since tN = T , we deduce that EP (LT ) = 1, which implies the martingale property of (Lt )
as proved at the beginning of the proof of the Girsanov Theorem (the process (Lt ) is a
non negative local martingale, thus a super-martingale, and has constant expected value).
2
The following results extends the Novikov theorem 3.4 for drift coefficients which are
functions of the Brownian motion. In that case, it is enough to impose a linear growth
condition on the coefficient b(t, x).
Proposition 3.6 (Beneš) Let B denote a standard d-dimensional Brownian motion, T > 0
and b : [0, T ] × Rd → Rd be a Borel function such that for some constant K
sup kb(t, x)k ≤ K(1 + kxk)
t∈[0,T ]
R
Rt
t
holds for any x ∈ Rd . Then if Lt = exp 0 b(s, Bs )dBs − 21 0 kb(s, Bs )k2 ds , one has
E(LT ) = 1 and the process (Lt , 0 ≤ t ≤ T ) is an (Ft )-martingale.
Proof. Let N and for n ∈ {0, · · · , N}, let tn =
T
.
N
For every n ∈ {1, · · · , N},
T
2
kb(s, Bs )k ds ≤ 2K
1 + sup kBs k .
N
0≤s≤T
tn−1
Z
tn
2
2
For any constant c > 0, the function x → exp[c(1 + kxk2 )] is convex. Hence Yt = exp c(1 +
2 kBt k2 ) is a submartingale and for C small enough (such that CT < 21 ) one has EP eC(1+kBT k ) <
+∞. Using Doob’s inequality, we deduce that
i
h
i
h
C
2
2) 2
(1+kBk
C(1+sup0≤t≤T kBt k2 )
t
≤ 4 EP eC(1+kBT k ) < +∞.
= EP sup e 2
EP e
0≤t≤T
Proposition 3.5 concludes the proof, choosing the instants tn such that ∆t = tn − tn−1 =
2C < T1 .
2
Rt
Example 3.7 Choosing b(t, x) = λx and using the identity Bt2 = 2 0 Bs dBs + t (which a
straightforward consequence of Itô’s formula) we obtain that by Proposition 3.6 the process
2
Z
Bt − t λ2 t 2
t → exp λ
−
B ds
2
2 0 s
is an (FtB )-martingale.
In dimension 1, one has other criteria which replace the Novikov condition. The proof
can be found in [6].
Proposition 3.8 (Kazamaki’ theorem) Let (θt ) be a local martingale such that the process
1
sub-martingale. The Doléans-Dade exponential Zt (θ) =
t →he 2 θt is a uniformly integrable
i
R
Rt
1 t 2
exp 0 θs dBs − 2 0 θs ds is a uniformly integrable martingale.
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Existence of weak solutions
3.5 Existence of weak solutions
The Girsanov Theorem proves the existence of weak solutions for some SDEs which do
not have strong solutions. The change of probability if provided by means of a generalization
of the Novikov condition.
Proposition 3.9 Let B denote a standard d-dimensional Brownian motion, T > 0 and
b : [0, T ] × Ω → Rd denote a Borel function such that there exists a constant K such that
x ∈ Rd
sup kb(t, x)k ≤ K(1 + kxk).
(3.5)
t∈[0,T ]
for any x ∈ Rd . Then for any x ∈ Rd , the SDE
Z t
Xt = x + Bt +
b(s, Xs )ds , 0 ≤ t ≤ T
(3.6)
0
(i)
(i)
(i)
(i)
has a weak solution on the interval [0, T ]. Furthermore,
if (Ω(i) , F (i) , (F
R
t ), P , B , X ),
T
(i)
i = 1, 2 are weak solutions, such that P (i) 0 kb(t, Xt )k2 dt < +∞ = 1, the both pairs
(B (i) , X (i) ) have the same distribution on this respective probability spaces.
Proof. Let us prove the existence of a weak solution. Let (Ω, F , (Ft), P) be a probability
space endowed with a filtration and let B be a (Ft ) Brownian motion. The growth conditions
imposed on b and Proposition 3.6 prove that for any x ∈ Rd , the process
!
Z t
d Z t
X
1
Zt = exp
bi (s, x + Bs )dBsi −
kb(s, x + Bs )k2 ds , 0 ≤ t ≤ T,
2
0
i=1 0
is a (true) P-martingale. This yields that the measure Q with density ZT with respect to P
is a probability, and that the process
Z t
Wt = Bt −
b(s, x + Bs )ds , 0 ≤ t ≤ T,
0
is a Q Brownian motion (with W0 = 0).
R t Therefore, under the probability Q, the process
Xt = x + Bt satisfies Xt = x + Wt + 0 b(s, Xs )ds with a Q-Brownian motion W . Hence
(X, W, Q) is a weak solution to (3.6) with the initial filtration.
The proof of uniqueness can be read in [7], page 304.
2
The following result extends the Beneš Theorem (Proposition 3.6) to drifts which are
functions of more general processes. One needs to control both the growth and the increments.
Proposition 3.10 (Non explosion criterion) Let T > 0, θt = b(t, Bs ) where b : [0, +∞[×R →
R is a function such that
|b(t, x) − b(t, y)| ≤ K|x − y| , ∀x, y ∈ R , ∀t ≥ 0,
supt≤T |b(t, 0)| ≤ K
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for some constant K. Then the process (Zt , t ∈ [0, T ]) defined by
Z t
Z
1 t
2
Zt (θ) = exp
b(s, Bs )dBs −
b(s, Bs ) ds
2 0
0
is an (Ft )-martingale. More generally, the process t 7→ Zt (θ) is an (Ft )-martingale as soon
as the SDE
dXt = dBt + b(t, Xt )dt , X0 = 0
admits a unique weak solution on the whole half-line [0, +∞[ (that is when there is no
explosion).
3.6 Examples of applications to computations of expectation
The Girsanov Theorem provides an efficient tool to compute expected values of some
functionals of Brownian motion. We will describe this in three cases.
1) We want to compute
Z T
I = EP Bt exp
θs dBs ,
0
for some t < T and some deterministic
function θ in L2 ([0, T ]). Introduce the density
R
RT
T
LT = exp 0 θs dBs − 12 0 θs2 ds , and then the probability Q with density LT with respect
to P on FTB . (One sees easily that
R the NovikovR condition
is satisfied.) Then, under P, the
t
1 t 2
martingale property of Lt = exp 0 θs dBs − 2 0 θs ds for the natural Brownian filtration
Rt
and the Girsanov Theorem prove that for Wt = Bt − 0 θs ds,
Z t
Z t
EP (Bt LT ) = EP (Bt Lt ) = EQ (Bt ) = EQ Wt +
θs ds =
θs ds.
0
0
RT
Rt
Hence I = 0 θs ds exp 12 0 θs2 ds . In the particular case θ = 1, the previous computa
t
tions show that EP Bt eBt = t e 2 (which can of course be proved directly by means of the
distribution of Bt ).
2) Let us go back to the computation of the value of a European call from section 2.6.2.
Recall that we have to compute explicitly
i
h
r(t−T ) et,x
+
C(t, x) = E e
(ST − K) ,
where SeTt,x = xeσ(BT −Bt )+
BT −t , so that
2
r− σ2
(T −t)
"
. The distribution of BT − Bt is the same as that of
r(t−T )
C(t, x) = EP e
x eσBT −t +
2
r− σ2
(T −t)
−K
+ #
i
h
σ2
= EP xeσBT −t − 2 (T −t) 1 σBT −t +(r− σ2 )(T −t)
2
{xe
>K}
σ2
− Ker(t−T ) P xeσBT −t +(r− 2 )(T −t) > K .
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Examples of applications to computations of expectation
63
Let us introduce the probability density
Z T −t
Z
1 T −t 2
1 2
LT −t = exp
σdBs −
σ ds = exp σBT −t − σ (T − t) .
2 0
2
0
Let Q denote the probability with density LT −t with respect to P. The process Ws = Bs −σs
is an (FsB )-Brownian motion under Q. Furthermore, since Bs = Ws + σs, we have


2
σ
(T − t) > ln(K/x) .
EP xLT −t 1( σB +(r− σ2 )(T −t) )  = xQ σWT −t + r +
2
2
>K
xe T −t
The other term can we written in a similar way. Indeed, it is equal to
σ2
−r(T −t)
(T − t) > ln(K/x) .
Ke
P σBT −t + r −
2
This yields immediately the formula obtained for C(t, x) in section 2.6.2 in terms of the
repartition function F of a standard Gaussian random variable and of reals d1 and d2 ,
C(t, x) = xF (d1 ) − Ke−r(T −t) F (d2 ),
r(T −t) √
1
xe
1 2
d1 = √
ln
+ σ (T − t) and d2 = d1 − σ T − t ;
K
2
σ T −t
note that the relation between d1 and d2 is obvious.
Let us notice an important duality formula between the put and the call. The dual put
to C is defined for t ∈ [0, T ] by
h
+ i
P (t, x) = E er(t−T ) K − S̃Tt,x
.
Clearly,
h
i
C(t, x) − P (t, x) = er(t−T ) E(S̃Tt,x ) − K = x − K e−r(T −t) .
It is often better to compute the put (using simulations and the de Monte-Carlo method)
because its variance is smaller that that of the call, which gives a better accuracy of the
approximation for a given number of simulations. The value of the call is then easy to deduce
because of the duality formula.. The value of the put can of course be computed in a similar
way to that of the call, using the Girsanov Theorem.
In this model, if the price of the risky asset is divided by that of the risky asset at time
t, that is if one wants to study the ratio
−rt
S̄t = e
”
“
2
σBt + b−r− σ2 t
St = e
,
then the Girsanov Theorem shows that for λ = b−r
, at any time T > 0 choosing the
σ
probability Q with density
λ2
LT = exp −λBT − T
2
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The Girsanov Theorem
with respect to P, then the process (Wt = Bt +λt, 0 ≤ t ≤ T ) is a (Ft , 0 ≤ t ≤ T )-martingale
2
under Q. For any t ≤ T ons has S̄t = exp(σWt − σ2 t). Therefore, under Q, (S̄t , 0 ≤ t ≤ T )
est is a (Ft , 0 ≤ t ≤ T )-martingale. One says that Q is the neutral risk probability.
3) Finally, to compute
let for t ∈ [0, T ]
h
i
R
λ2 T
2
I = EP Φ(BT )e− 2 0 Bs ds ,
Lt = exp −λ
Z
t
0
λ2
Bs dBs −
2
Z
0
t
Bs2 ds
.
The Beneš criterion (Proposition 3.6) or the non-explosion criterion (Proposition 3.10) applied to the function b(s, x) = −λx shows that (LtR, 0 ≤ t ≤ T ) is a (Ft )-martingale.
t
Furthermore, Itô’s formula (Example 3.7) shows that 0 Bs dBs = 21 (Bt2 − t), that is
Z
Bt2 − t λ2 t 2
Lt = exp −λ
−
B ds .
2
2 0 s
Rt
Furthermore, under the probability Q, Wt = Bt + λ 0 Bs ds is a Brownian, that is under Q,
Bt = Wt − λ
Z
t
Bs ds
0
is an Ornstein-Uhlenbeck
process. Using exercise 2.2, we know that
under Q the random
Rt
1
(1 − e−2λt ) distribution. Thus,
variable Bt = e−λt 0 eλs dWs et has a Gaussian N 0, 2λ
2
B 2 −T
B −T
λ T2
λ T2
I = EP LT e
Φ(BT ) = EQ
Φ(BT ) ,
and this last integral can be explicitly computed by means of the Gaussian distribution of
BT under Q.
3.7 The predictable representation theorem
In this section, B will denote a standard Brownian motion and (FtB ) its natural filtration (completed). This assumption is crucial. We will only use one probability P and let E
denote the expected value under P. In that case, we will prove that any continuous (FtB )
local martingale can be written as the stochastic integral of a process in H2loc .
Let us at first represent square integrable, FT -measurable random variables.
We at first prove some technical lemma about a family of random variables dense in
2
L (FTB ), which explains how the measurability wit respect to the σ algebra FTB is used.
Lemma 3.11 For T > 0, the family of random variables {Ψ(Bt1 , · · · , Btn ) : 0 < t1 <
· · · tn ≤ T, ψ ∈ C0∞ } is dense in L2 (FTB ).
Proof. Let (tk , k ∈ N) denote the sequence of elements of the countable set Q ∩ [0, T ] and
for each integer n ≥ 1, let Bn denote the σ-algebra generated by the random variables
Btk , 1 ≤ k ≤ n. The sequence (Bn ) is a countable filtration and the σ-algebra B∞ =
σ(Bti , i ≥ 1) is equal to FTB . Furthermore, given any random variable X ∈ L2 (FTB ), the
sequence Xn = E(X | Bn ), n ≥ 1 is a discrete martingale which converges a.s. and in L2
Stochastic Calculus 2 - Annie Millet
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The predictable representation theorem
65
to X. By definition of Bn , each random variable Xn can be written as gn (Bt1 , · · · , Btn ),
where gn is Borel. Finally, for any ǫ > 0, the random variable gn (Bt1 , · · · , Btn ) can be
approximated in L2 (dP) up to ǫ by a random variable Ψn (Bt1 , · · · , Btn ), where Ψn is of class
C ∞ with compact support.
2
Theorem 3.12 Let Z denote a square integrable, FTB -measurable random variable. There
exists a unique process (Ht , 0 ≤ t ≤ T ) in H2T such that
Z T
Z = E(Z) +
Hs dBs .
(3.7)
0
Proof. The fact that
Zt =
R t H is unique comes from the uniqueness of the Itô process
B
T
E(Z|Ft ) = E(Z) + 0 Hs dBs pour 0 ≤ t ≤ T (which is a martingale if H ∈ H2 ).
The proof of existence consists in three steps.
RT
(i) Let H = { 0 Hs dBs : H ∈ H2T }. This is a linear subspace of L2 (FTB ), with centered
elements, and it is isometric to the set H2T which is closed for the L2 ([0, T ] × Ω)-norm and
the measure dλ ⊗ dP. Hence H is a closed subset of L2 (FTB ). For any Z ∈ L2 (FTB ), let
RT
Hs dBs denote the orthogonal projection of Z − E(Z) on H. Then,
0
Z T
e
Z = E(Z) +
Hs dBs + Z,
0
where Ze is a square integrable, centered random variable which is orthogonal to H.
(ii) We prove that Ze is orthogonal to the set
Z T
Z T
1
2
2
E = exp
f (s)dBs −
f (s) ds : f ∈ L ([0, T ]) .
0
0 2
RT
RT
Let f ∈ L2 ([0, T ]) ; then 0 f (s)dBs is a Gaussian random
variable
with
N
(0,
f (s)2 ds)
0
R
RT
T
distribution and if X = exp 0 f (s)dBs − 21 0 f (s)2 ds , then E(X) = 1. (Note that This
is a Girsanov density and that the Novikov condition is satisfied). Its remains to check that
for X ∈ E, on can deduce that X − 1 ∈ H. Let X ∈ E and for t ∈ [0, T ], set
Z
Z t
1 t
f (s)2 ds , and Mt = exp(ξt ).
ξt =
f (s)dBs −
2
0
0
Then X = MT = exp(ξT ).
Itô’s formula shows that this Girsanov density satisfies the SDE dMt = Mt dξt + 12 Mt f (t)2 dt =
Mt f (t)dBt . This yields
Z T
X = MT = 1 +
Mt f (t)dBt
0
H2T .
and it suffices to check that t → Mt f (t) ∈
Fubini’s theorem implies that
Z T
Z T
E
Mt2 f (t)2 dt =
E(Mt2 )f (t)2 dt
0
0
Z T h R
i Rt 2
R
t
1 t
2
=
E e 0 2f (s)dBs − 2 0 4f (s) ds e 0 f (s) ds f (t)2 dt
0
Z T R
t
2
=
e 0 f (s) ds f (t)2 dt.
0
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66
Since
3
kf k22
=
RT
0
2
f (s) ds, we deduce E
R
T
Mt2 f (t)2 dt
0
The Girsanov Theorem
≤ kf k22 exp(kf k22 ) < +∞.
(iii) We now prove that any random variable X orthogonal to E is a.s. equal to 0, which
implies that the vector space generated by E is dense in L2 (FTB ).
Let X ∈ L2 (FhTB ) be such that E(XY
i ) = 0 for all Y ∈ E. then, given any function
RT
2
f ∈ L ([0, T ]), E X exp 0 f (s)dBs
= 0. Let t0 = 0 < t1 < · · · < tn ≤ T denote a
Pn
subdivision and let f = j=1 aj 1[0,tj ] be a step function. For any choice of the subdivision
and of the constants aj we deduce
!#
"
n
X
aj Btj
= 0, ∀a ∈ Rn .
Φ(a) = E X exp
j=1
Using analytic continuation, we deduce that Φ(a) = 0 for any a ∈ Cn .
Let Ψ be a function
of class C ∞ with compact support, and let Ψ̂ be its Fourier transform
R
defined by Ψ̂(y) = Rn Ψ(x) exp[i(x, y)]dx. The Fourier inversion formula proves that Ψ(x) =
R
(2π)−n Rn Ψ̂(y) exp[−i(x, y)]dy. Therefore, the Fubini Theorem yields
E [XΨ(Bt1 , · · · , Btn )] = (2π)
−n
= (2π)−n
Z
ZR
n
−i[y1 Bt1 (ω)+···+yn Btn (ω)]
dy
Ψ̂(y)E X(ω)e
Ψ̂(y)Φ(−iy)dy = 0.
Rn
Finally, Lemma 3.11 concludes the proof.
2
The following theorem will be very important in finance. It allows to build hedging
portfolios.
Theorem 3.13 (Brownian martingale representation theorem) Let B denote s standard
Brownian motion, (FtB ) its natural completes filtration, M be a (FtB ) local martingale. Then
there exists x ∈ R and θ ∈ H2loc such that
Mt = x +
Z
t
θs dBs .
0
If M is a ”true” square integrable (FtB )-martingale, then θ ∈ H2 .
Proof. Let M be a square integrable such that M0 = 0. Fix T > 0 ; using RTheorem 3.12, we
T
deduce that there exists a process (θtT , 0 ≤ t ≤ T ) ∈ H2T such that MT = 0 θtT dBt a.s. The
′
uniqueness proves that for T < T ′ , the processes θT and θT are equal dt ⊗ dP-a.e. on the
set [0, T ] × Ω. Therefore, one can define almost surely a process θ by the formula : Ht = HtT
RT
for all 0 ≤ t ≤ T . This implies that θ. 1[0,T ] (.) belongs to H2T and that MT = 0 θs dBs .
The result about (FtB )- local martingales is admitted. Notice that in that case, local
martingales are automatically continuous.
2
Finally, in some cases the Clark-Ocone formula - see e.g. [9] - provides an explicit formulation on the process θ.
Stochastic Calculus 2 - Annie Millet
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67
Exercises
3.8 Exercises
Exercise 3.1 Let B be a one-dimensional Brownian motion, α and β be non negative
constants. One wants to compute
Z
β2 t 2
2
I = E −αBt −
B ds .
2 0 s
1. Prove that the process defined by
Z
β 2
β2 t 2
β
Lt = exp − (Bt − t) −
B ds
2
2 0 s
is a (Ft )-martingale under P.
2. Let Q denote the probability defined by dQ|Ft = Lβt dP|Ft . Prove that under Q, Bt =
Rt
Wt − β 0 Bs ds where W is a Brownian motion.
3. Deduce that
β 2
2
I = EQ exp(−αBt + (Bt − t)) .
2
− 21
.
sinh(βt)
Using Exercise 2.2, deduce that I = cosh(βt) + 2α
β
4. Deduce that for every λ > 0, one has
Z t
− 21
√
2
EP exp −λ
Bs ds
= cosh( 2λt)
.
0
Exercise 3.2 Let (ht ) and (Ht ) be r-dimensional,
processesP
and let
R t Bk be ka
Pr (FRt )-adapted
t k
r
k
r-dimensional (Ft )-Brownian motion. Let Mt = k=1 0 hs dBs and Nt = k=1 0 Hs dBs .
Suppose that h satisfies
the Novikov condition and let Q denote the probability with density
1
exp Mt − 2 hMit with respect to P on Ft . Prove that the process (Nt − hM, Nit ) is an (Ft )
local martingale under Q. Under which condition is it a martingale under Q ?
Exercise 3.3 Let α, β and γ denote real constants.
R
R
t Bs
αBt t βBs
1. Compute E 0 e ds , and then E e
e ds .
0
R t B +γs
2. Set A(t, γ) = 0 e s
ds. Compute E(A(t,
γ)). Using Girsanov’s theorem, prove that
R
αBt t βBs
E(A(t, γ)) gives a way to compute E e
e ds .
0
Exercise 3.4 Let (Bt ) be a one-dimensional Brownian motion, α, β and σ beR bounded
t
deterministic functions from R to R. Suppose that σ does not vanish and set b(t) = 0 β(s)ds.
Finally let (rt ) denote the solution to the SDE
drt = σ(t) dBt + [α(t) − β(t) rt ] dt , r0 ∈ R.
1. Prove that
−b(t)
rt = e
r0 +
Z
t
b(u)
σ(u)e
dBu +
0
Z
0
t
b(u)
α(u)e
du .
2. Compute E(rt ) and Cov(rs , rt ) for 0 ≤ s ≤ t.
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3
The Girsanov Theorem
; suppose that θ is bounded and let
3. Let θ(s) = − α(s)
σ(s)
Lt = exp
Z
t
0
1
θ(s)dBs −
2
Z
t
2
θ(s) ds .
0
Let QR1 denote the measure with density Lt with respect to P on Ft and let Wt1 =
t
Bt − 0 θ(s)ds. Prove that (rt eb(t) , t ≥ 0) is a Q1 (Ft )-martingale.
dWt1 et Z0 = 1. Let Q2 be the
4. Let (Zt ) denote the solution to the SDE dZt = Zt β(t)
σ(t)
probability with density Zt with respect to P on Ft . Prove that (rt ) is a (Ft )-martingale
under Q2 .
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4 Applications to finance
Conventions of notations
Let (Ω, F , P) be a probability space, (Bt , 0 ≤ t ≤ T ) a standard k dimensional Brownian
standard motion and (Ft , 0 ≤ t ≤ T ) its natural completed filtration.
4.1 Continuous financial market
We consider a more general version than the model described in section 2.6.2.
4.1.1 Financial market with d risky assets and k factors
We consider a financial market where the assets are traded continuously. Some of these
assets, called risky assets, are random and depend on a k dimensional Brownian motion B.
The market has d + 1 basic assets (Sti , 0 ≤ i ≤ d, t ≥ 0) and other contingent claims,
that is derived financial products. We suppose that non of the assets is paying dividends,
that there is no transaction cost, that one can buy and sell continuously and the assets are
infinitely divisible (their value is a real number). Finally, we suppose that the price of the
asset number i at any time, say Sti , Itô process constructed in terms of B and taking only
(strictly) positive values. Let St = (St0 , · · · , Std ) for any t ≥ 0. All the results would extend
to continuous almost surely positive continuous semi-martingales.
Example 4.1 The Black & Sholes model For example in the Black & Sholes model
described in section 2.6.2, the asset S 0 is non-risky such that dSt0 = rS 0 dt and S00 = 1,
that is to say St0 = ert . One also has a risky asset which is a geometric Brownian motion
satisfying dSt1 = σSt1 dBt + bSt1 dt. If S01 > 0, then we have St1 > 0 a.s. for all t ≥ 0.
In the sequel, we will see that the generalized Black & Sholes model
R where
the non
t
loc
0
0
risky asset has a rate rt ∈ H1 , that if S0 = 1, we still have St = exp 0 rs ds and that
the d risky assets S i , 1 ≤ i ≤ d are strictly positive valued general Itô processes. The
stochastic volatility at time t is the ratio between the diffusion coefficient and the value Sti
and similarly, we let bi (t) denote the ratio between the drift coefficient and the value, that
is for i = 1, · · · , k,
k
X
i
dSs =
σji (s)Ssi dBsj + bi (s)Ssi ds.
j=1
We will give later sufficient conditions on the coefficients which ensure the existence and
uniqueness of the solution to such SDEs.
Definition 4.2 A numéraire is an Itô process (St ) such that St > 0 a.s. for any t ≥ 0.
We use the non-risky asset S 0 in the generalized Black & Sholes model as a numéraire
with 1 as monetary unit at time 0, that is S00= 1. Atany time t > 0, the value of the risky
Rt
assets is computed with respect to St0 = exp 0 rs ds , which means that their behavior is
compared with that of the non-risky one which is taken to be 1. Using this « new monetary
unit », the value of another assets Xt is
Z t
Xt
X̃t = 0 = Xt exp −
rs ds .
St
0
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4
Applications to finance
We also have « contingents claims » or « derived products » whose price’s evolution
depends on that of the basic assets S i , 0 ≤ i ≤ d. More precisely, we have a fixed time T > 0
called the « maturity » of the contingent claim and its value ζ at time T which may depend
on all the trajectories of the basic assets (Sti , i = 0, · · · , d, 0 ≤ t ≤ T ). Hence the value of
the claim at time T is a random variable which is FT -measurable. For a European call in
the Black & Sholes, we have ζ = (ST1 − K)+ .
We want to « duplicate » the contingent claim ζ. We may use some initial amount which
at any time t is split among the various basic assets without adding or subtracting any
money, so that at the terminal time T , the sum of all the amounts invested in the assets is
equal to ζ. The aim is to find the initial investment at time 0 and the way is should be split
among the various assets at any time t ∈ (0, T ].
4.1.2 Description of the strategies
Let us give a formal definition of duplication.
Definition 4.3 A (trading) strategy is a process Θ = (θs0 , · · · , θsd ), s ∈ [0, T ]) such that
Rt
each component θi is (Ft )-progressively measurable and 0 θsi dSsi exists for all i = 0, · · · , d.
In the Black & Sholes model with two assets with constant coefficients, since the processes
S 0 and S 1 have continuous trajectories, that is a.s. bounded on [0, T ), one may only impose
0
loc
1
loc
the coefficients
case the deterministic integrals
Rthat
R t 1 1 θ ∈ H1 and θ ∈ H2 . Indeed,
R t in that
t 0 rs
1 1
θ e ds, 0 θs bSs ds and the stochastic integral 0 σθs Ss dBs are well defined. Note that
0 s
the last one is a local martingale.
Let us recall the assumptions made on the market (and described in section 2.6.2) :
– There is no transaction cost when one buys or sells assets.
– One allows unlimited uncovered selling, that is that the processes θti may take on
negative values and are not lower bounded.
– The assets are indefinitely divisible, that means that one may sell or buy an arbitrary
proportion of an asset ; the processes θi can take all real values.
– « Trading » takes place continuously, that means that one may sell or buy at any non
negative time t ∈ [0, T ].
The value of the portfolio VtΘ associated with the strategy Θ at time t is :
VtΘ
= (θt , St ) =
d
X
θti Sti .
(4.1)
i=0
The fact that no money is added or withdrawn is modeled by the following definition :
Definition 4.4 A strategy Θ is self-financing (or
the
corresponding portfolio V Θ is selfR
P
t
d
financing) if at any time t ∈ [0, T ], VtΘ = V0Θ + i=0 0 θsi dSsi a.s.
The variations of the value of the portfolio only come from the way the money is invested
on the various basic as well as of the changes of the values of these assets.
In the sequel, we will change the numéraire in such a way that the processes which describe the prices of the risky assets after the unit change become martingales. The following
result proves that the self-financing property is not changed by this transformation.
Si
Proposition 4.5 Let (Xt , t ∈ [0, T ]) be a numéraire, Xtt denote the process of the risky
asset after changing the numéraire. If the strategy Θ is self-financing, the portfolio after
VΘ
changing the numéraire, that is Xt t , remains self-financing.
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Continuous financial market
Proof. Since X is a strictly positive Itô process, Y = X1 is also a strictly positive Itô process
and the Itô formula implies that for any i = 0, · · · , d,
i
St
d
= d(StiYt ) = Sti dYt + Yt dSti + dhS i, Y it .
Xt
P
If the process V Θ is self-financing, dVtΘ = di=0 θti dSti a.s. and V Θ is an Itô process. Applying
once more Itô’s formula yields
Θ
Vt
d
= VtΘ dYt + Yt dVtΘ + dhV Θ , Y it
Xt
i
d
d
X
X
i i
St
i
i
i i
i
.
2
=
θt St dYt + Yt θt dSt + dhθ S , Y it =
θt d
X
t
i=0
i=0
The following notion is also preserved by a change of numéraire.
Definition 4.6 A strategy Θ has non-negative if it is self-financing and if for every t ∈
[0, T ], VtΘ ≥ 0.
When taking the non-risky asset S 0 as the numéraire, after this change of unit the prices
of the basic risky assets are S̃t = (1, S̃t1 , · · · , S̃td ). They are again Itô processes and a portfolio
is self-financing if and only if after change of numéraire one has
dṼtΘ
=
d
X
i=1
θti dS̃ti, ∀t ∈ [0, T ].
One deduces that a self-financing portfolio is completely determined by its initial value V0Θ
and the amounts inverted on each risky asset at any time, that is the processes (θti , i =
1, · · · , d, t ∈ [0, T ]). Indeed, dS̃t0 = 0, Ṽ0Θ = V0Θ and
ṼtΘ
=
V0Θ
+
d Z
X
i=1
t
0
θsi dS̃si
and
θt0 can be deduced by ṼtΘ = θt0 +
d
X
θti S̃ti .
i=1
Since after changing the monetary unit the price of the non-risky asset remains constant,
it is a (trivial) martingale. One would like to turn the other assets into martingales. The
preceding chapter has shown that this is possible by means of a change of probability which
can cancel the drift term.
4.1.3 Arbitrage-free condition
This justifies the following definition of admissible strategy.
Definition 4.7 (i) An equivalent martingale measure is a probability Q equivalent with P
and such that under Q the prices S̃ i , i = 1, · · · , d after the change of numéraire are (Ft )martingales.
More precisely, we assume that there exists a Q Brownian motion W such that
P
dS̃ti = kj=1 Hji (t)dWtj .
(ii) Let Q be an equivalent martingale measure. The strategy Θ is Q-admissible if it is
self-financing and if (ṼtΘ , 0 ≤ t ≤ T ) is a (Ft , 0 ≤ t ≤ T ) martingale under the probability
Q.
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Applications to finance
The following notion is fundamental ; it reflects the fact that the market is operating
properly and that there is no « free lunch », that is that one cannot gain a strictly positive
amount at time T with no investment at time 0. The opposite property is described in the
following definition.
Definition 4.8 (Arbitrage possibility) A self-financing strategy Θ is an arbitrage possibility
if
(i) V0Θ = 0 et VTΘ ≥ 0 a.s.
(ii) P(VTΘ > 0) > 0.
A correctly operating financial market must satisfy the no-arbitrage property, that is
does not contain an arbitrage possibility ; one also says that it is arbitrage-free. Since two
equivalent measures have the same null sets, in the arbitrage possibility one may replace
the probability P by an equivalent equivalent measure Q.
The following result proves that the existence of an equivalent martingale measure Q
excludes any arbitrage possibility when restricted to Q-admissible strategies or to strategies
with non-negative wealth. We will see that in the case of the Black & Sholes mode, this last
property is equivalent with the no arbitrage property.
Theorem 4.9 Let Q be an equivalent martingale measure ; then :
(i) The Q- admissible strategies Θ are arbitrage-free.
(ii) The strategies Θ with non-negative wealth are arbitrage-free.
Proof.
(i) Let Θ be an admissible strategy. Then V0Θ = Ṽ0Θ = EQ (ṼTΘ | F0) = EQ (ṼTΘ ). If
Θ
ṼT ≥ 0, Q a.s. and if P(VTΘ > 0) > 0, then we deduce that Q(ṼTΘ > 0) > 0 and hence that
EQ (ṼTΘ ) > 0. This forbids V0Θ = 0.
P
(ii) Let Θ be a strategy with non-negative wealth ; let dS̃ti = kj=1 Hji (t)dWtj for t ∈ [0, T ],
where (Wt , 0 ≤ t ≤
is a standard k-dimensional Q Brownian motion. Then for any
PT ) P
t ∈ [0, T ], dṼtΘ = di=1 kj=1 θti Hji (t)dWtj Q a.s. One deduces that Ṽ Θ is a non-negative
(Ft ) local martingale under Q. Hence it is a super-martingale. Therefore, for any t ∈ [0, T ],
EQ (ṼtΘ |F0) ≤ Ṽ0Θ = V0Θ Q a.s. Hence if Θ is an arbitrage possibility, EQ (ṼTΘ ) ≤ 0. On the
other hand, since ṼTΘ ≥ 0 P a.s., and hence Q a.s., we deduce ṼTΘ = 0 Q a.s., hence P a.s.,
which provides a contradiction, which concludes the proof.
2
One should be aware of a difficulty arising in continuous time : the existence of an
equivalent martingale measure does not imply that the no-arbitrage property is full-filed
without any restriction on the strategies. Indeed, let Z be a FTW -measurable, square integrable random variable with zero mean under the probability Q and assume that Z is not
identically null. Then Theorem 3.12 proves that there exists H ∈ H2T (under Q) such that
RT
Z = 0 Ht dWt . In the Black & Sholes model, this implies the existence of a self-financing
strategy Θ = (θt0 , θt1 ) such that Ṽ0Θ = 0 et ṼTΘ = Z.
4.1.4 Neutral risk probability
In this section, we assume that there exists a martingale measure Q equivalent
to P.
Definition 4.10 One says that a pay-off given at the terminal time T by an FT -measurable
random variable ζ is Q-replicable if there exists a Q-admissible strategy Θ such that ζ is the
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73
Continuous financial market
terminal value of the associated portfolio, that is if ζ = VTΘ . In that case, one says that the
strategy Θ replicates the pay-off at terminal time T , or is « replicating ».
Intuitively, this means that the portfolio is replicating the pay-off « in all the states of the
world » that is for P (or Q) every ω.
As an exercise, one can prove that if a pay-off is replicable at time T , it remains replicable
after a change of numéraire with the same strategy.
The following result proves that at any time t ≤ T , the value of the portfolio does not
depend on the replicating strategy (but it may depend on Q).
Theorem 4.11 Let T > 0, ζ a Q-replicable contingent claim (FT -measurable random variable). Let Θ be a replicating strategy and let V Θ be the corresponding portfolio ; then for
any time t ∈ [0, T ], VtΘ = St0 EQ (ṼTΘ |Ft ) does not depend on the replicating strategy. It is
called the price of the contingent claim at time t.
Proof. For any t ∈ [0, T ], ṼtΘ = EQ (ṼTΘ |Ft ), that is
ζ Θ
0
Vt = St EQ
Ft ,
ST0
does not depend on the strategy Θ.
2
Obviously, if a Q-admissible strategy Θ replicates a contingent claim ζ ≥ 0, then it has
non-negative wealth. However, a non-negative wealth strategy does not determine uniquely
the price of the contingent claim. Indeed, Harrison & Pliska have proved the existence of a
strategy Φ with non-negative wealth such that V0Φ = 1 and VTΦ = 0 : this strategy is called
the « suicide strategy ». Clearly, if the strategy Θ replicates the claim ζ, so does the strategy
Θ + Φ.
Going back to the Black & Sholes model, we have seen at the end of the previous chapter
2
, then the probability Q with de density exp(−λBT − λ2 T ) with respect to P
that if λ = b−r
σ
has the following property : under Q the process (Wt = Bt + λt, 0 ≤ t ≤ T ) is a Brownian
motion and (S̃t = e−rt St , t ≥ 0) is a Q-martingale. The above computation shows that under
Q, if ζ is a FT -measurable random variable, then
VtΘ = e−r(T −t) EQ (ζ|Ft) = ert EQ (e−rT ξ|Ft).
Under Q, the conditional expectation EQ (ζ|Ft) can be deduced from the corresponding one
under P multiplying by the exponential factor exp(−r(T −t)), that is using the actualization
factor r. The diffusion coefficients no longer appear in the coefficients of the diffusion S, as
if the traders were neutral with respect to risk. The probability Q is called the neutralrisk probability. The starting probability P is called the « historical » or « objective »
probability.
The remaining problem is that of replicating a contingent claim.
Definition 4.12 (i) The market is complete if at any time T , every pay-off is replicable.
(ii) Let Q be an equivalent martingale measure. One says that the market is Q-complete
if for every FT -measurable random variable ζ such that Sζ0 ∈ L1 (Q), a claim with terminal
T
value ζ is Q-replicable.
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Applications to finance
Note that if the market is arbitrage-free, there is at most an equivalent martingale
measure Q such that the market is Q-complete.
Theorem 4.13 Let the market be complete. Let Q1 and Q2 denote two equivalent martingale
measures such that the market is Qi -complete, i = 1, 2 ; then Q1 |FT = Q2 |FT .
Proof. Let A ∈ FT . Then 1A ∈ L1 (Q) and ST0 1A is FT -measurable and is the terminal value
of a Qi -admissible trading strategy Θi , i = 1, 2. We deduce that
i
V0Θ
=
i
Ṽ0Θ
=
i
EQi (ṼTΘ )
= EQi
ST0 1A
ST0
= Qi (A).
Using Theorem 4.11, we deduce that the value of the replicating portfolio does not depend on
1
2
strategy ; at time 0, this describes the model is arbitrage-free. We deduce that V0Θ = V0Θ ,
and hence Q1 (A) = Q2 (A).
2
4.2 Extended Black & Sholes model
We at first generalize the Black & Sholes model in the case of one risky asset, which
has been described in the previous chapters, and prove that in a more general setting there
exists an equivalent martingale measure Q as soon as the model is arbitrage-free.
Furthermore, we give sufficient conditions for the market to be Q-complete and compute
the hedging strategy, that is the replicating strategy Θ and the price of the contingent claim
at every time.
R
t
Let r be a process in H1loc and St0 = exp 0 rs ds .
On the other hand, let S i , i = 1, · · · , d be risky assets which are Itô processes as follows :
Sti
=
S0i
+
k Z
X
j=1
0
t
σji (s)Ssi dBsk
+
Z
t
bi (s)S i(s)ds ,
(4.2)
0
where B is a k-dimensional Brownian motion, the processes (σji (s), 0 ≤ s ≤ T ) and
(bi (s), 0 ≤ s ≤ T ) are progressively measurable and such that for some constant M > 0 one
has sup0≤t≤T (kσ(t)k + kb(t)k) ≤ M. Then using exercise 2.4 for i = 1, · · · , d, we deduce
Sti
1 i
2
= exp
σ (s)dBs +
(b (s) − kσ (s)k )ds
2
0
0
"
# !
Z
Z
k
k
t
X t
X
1
:= exp
σji (s)dBsj +
bi (s) −
|σji (s)|2 ds .
2
0
j=1 0
j=1
Z
t
i
Z
t
i
(4.3)
Furthermore, equation (4.3) is also satisfied if the Itô processes (hence a.s. continuous proP
H i (t)
cesses) S i such that dSti = kj=1 Hki (s)dBsk + H0i (s)ds and such that if one lets σki (t) = Sk i
t
RT
H0i (t)
2
i
for i = 1, · · · , d and b (t) = S i(t) , one has 0 (kσ(s)k + kb(s)k)ds < +∞ a.s.
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4.2.1 Arbitrage-free and change of probability - Risk Premium
In the previous section, we have shown that the existence of an equivalent martingale measure Q implies the arbitrage-free condition among the strategies with non-negative
wealth. We show that the converse holds in this model.
Suppose that the arbitrage free condition holds for strategies with non-negative wealth.
To focus on the notion of risk premium for one factor, let us at first assume that k = 1
and d = 2. In that case, a self-financing portfolio is entirely defined by its initial value V0Θ
and by the amounts θti , i ∈ {1, 2}, t ∈ [0, T ] invested at any time t on both assets. For all
t ∈ [0, T ], set θt1 = σt2 St2 and
θt2 = −σt1 St1 . Since S i is continuous, the properties imposed on
R
t
σ i show that the integrals 0 θsi dSsi are well-defined and (θt1 , θt2 ) is a self-financing strategy.
Rt
For every t ∈ [0, T ], St0 = exp( 0 rs ds), and
Rt
dS̃ti = d e− 0 rs ds Sti = σti S̃ti dBt + (bit − rt )S̃tidt,
which implies (due to cancellation of the coefficients of dBt ) :
dṼtΘ = σt2 St2 dS̃t1 − σt1 St1 dS̃t2 = e−
Rt
0
rs ds
St1 St2 [σt2 (b1t − rt ) − σt1 (b2t − rt )]dt.
Since this process is non-risky, the arbitrage-free assumption shows that its instanteanuous
interest rate is equal to rt , which means that
it is null when the prices are written after a
R
− 0t rs ds 1 2 2 1
St St [σt (bt − rt ) − σt1 (b2t − rt )], and suppose
change of numéraire. Indeed, set φ(t) = e
that there exists a set A ⊂ [0, T ] × Ω such that dλ ⊗ P(A) > 0 and φ(t)(ω) 6= 0 on A. Let
s(t) denote the sign of φ(t) when φ(t) 6= 0 and for i = 1, 2 set
ψ i (t) = s(t)θi (t) on A and ψ i (t) = 0 on Ac .
Then the strategy Ψ = (ψ1 , ψ2 ) is such that
dṼtΨ = |φ(t)|dt on A,
dṼtΨ = 0 on Ac .
b1 −r
b2 −r
The strategy Ψ is an arbitrage possibility and hence we deduce that tσ1 t = tσ2 t := λt .
t
t
This ratio is denoted by λt and called the market price of the risk Bt at time t. It gives
a value to the arbitrage one has to do at time t between rate and risk in the composition of
the portfolio.
Let us come back to the general case. For any time t ∈ [0, T ], the matrix σ(t) defines
a linear application, still denoted by σ(t) : Rk → Rd , whose matrix in the canonical
basis
Pk
i
k
i
is σ(t), that is, for any x = (x1 , · · · , xk ) ∈ R and i = 1, · · · , d, (σ(t)x) = j=1 σj (t)xj .
We
for a vector λt = (λ1t , · · · , λkt ) such that for any i = 1, · · · , d, bi (t) − rt =
Pk are i looking
j
j
k
j=1 σj (t)λt , that is λt is the risk premium associated with the Brownian motion Bt at
time t. Identifying a vector Rk with the column matrix of its components in the canonical
basis and letting 1̄ = (1, · · · , 1) ∈ Rd , this is equivalent to solving the equation
b(t) − rt 1̄ = σ(t)λt .
The following theorem shows the existence of the vector λ of the market risk premiums.
Theorem 4.14 Suppose that the non-negative wealth strategies are arbitrage-free. Then
there exists a progressively measurable process λ : [0, T ]×Ω → Rk such that for all t ∈ [0, T ],
b(t) − rt 1̄ = σ(t)λt dt ⊗ dP almost everywhere on [0, T ] × Ω.
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Proof. We prove that b(t) − rt 1̄ ∈ Im(σ(t)) for dt ⊗ dP almost every (t, ω) ∈ [0, T ] × Ω. We
will suppose that the choice of λ can be done in a progressively measurable way.
This reduces to checking that b(t) − rt 1̄ is orthogonal to every element of the kernel
of σ(t)∗ : Rd → Rk which is the adjoint of the linear map σ(t) : Rk → Rd , associated
with the transposed matrix of σ(t). Indeed, the inclusion Im(σ(t)) ⊂ Ker(σ(t)∗ )⊥ is obvious
by definition of the adjoint map : for any y ∈ Rk and z ∈ Rd , (σ(t)y, z)Rd = (y, σ(t)∗z)Rk .
Furthermore, the classical relations between the dimensions of the kernel and image of linear
maps and of the orthogonal supplement of a linear subspace prove that dim Im(σ(t)) ≤
d − dim Ker (σ(t)∗ ) = dim Im (σ(t)∗ ) ; exchanging the roles of σ(t)∗ and σ(t) = (σ(t)∗ )∗ we
deduce that dim Im (σ(t)∗ ) ≤ dim Im(σ(t)), and hence dim Im (σ(t)) = dim Im (σ(t)∗ ) =
dim (Ker(σ(t)∗ ))⊥ , that is Im(σ(t)) = (Ker(σ(t)∗ ))⊥ by the above inclusion.
Suppose that b(t) − rt 1̄ ∈ Ker(σ(t)∗ )⊥ does not hold for pour dt ⊗ dP almost every
(t, ω) ∈ [0, T ]×Ω. One could choose in a predictable way a non-null vector x(t) ∈ Ker(σ(t)∗ )⊥
such that (b(t) − rt 1̄, x(t)) 6= 0 on a subset A ⊂ [0, T ] × Ω with dt ⊗ dP (A) > 0. One can
obviously impose kx(t)k = 1. Since the prices of the basic assets are strictly positive, the
i
allows to construct
process Θ : Ω × [0, T ] → Rd defined for i = 1, · · · , d by θti = xS(t)
i
t
a self-financing strategy. Indeed, the self-financing condition
to deduce θt0 and the
R t allows
measurability conditions and the existence of the integrals 0 θsi dSsi are full-filed. Then one
i
i i
has x(t) = (ΘS)t , where the vector
(ΘS)t is defined by the equalities (ΘS)t = θt St .
Set st = signe (b(t)−rt 1̄, x(t) , with the convention signe(y) = 1 if y ≥ 0 and signe(y) =
−1 otherwise. ThenP
we construct another self-financing strategy Φ by setting Φt = s(t)Θt .
For any j, we have di=1 θti Sti σji (t) = 0 and
"
# d
d
X
X
X
Rt
j
i i i
i i i
i
i i
Φ
− 0 rs ds
S̃t θt (b (t) − rt ) dt.
st θt St σj (t)dBt +
θt st St (b (t) − rt )dt = dṼt = e
i,j
i=1
i=1
Thus we deduce that dṼtΦ ≥ 0 with dṼtΦ > 0 dt ⊗ dP a.e. on A. Hence the strategy Φ has
RT
non-negative wealth and is such that V0Φ = 0 and E 0 dṼtΦ > 0. This implies that on a set
RT
with strictly positive probability, ṼTΦ = 0 dṼsΦ > 0. Thus Φ is an arbitrage possibility.
Since the model is arbitrage-free among strategies with non-negative wealth, the vector
b(t) − rt 1̄ belongs to the orthogonal supplement of the kernel of σ(t)∗ , that is to the image
of σ(t), dt ⊗ dP almost everywhere.
2
Finally, the following theorem proves that if the risk-premium vector λ has more properties, then there exists an equivalent martingale measure that can be described explicitly
in terms of λ has further properties. This extends what we have observed in the case of the
Black & Sholes model. Indeed, in that case the risk-premium λ is such that σλ = b − r, that
is λ = b−r
and the risk-neutral
probability
Q has been constructed using the exponential
σ
martingale Lt = exp −λBt −
λ2
t
2
.
Theorem 4.15 Suppose that there exists a risk-premium vector λ : [0, T ] × Ω → Rk , that
is a progressively measurable process such that b(t) − rt 1̄ = σ(t)λt and suppose furthermore
that :
(i) λ ∈ H2loc (Rk ).
R
Rt
t
(ii) For any t ∈ [0, T ], if one lets Lt = exp − 0 λs dBs − 21 0 kλs k2 ds one has
R
RT i
σs (dBs +λs ds)− 21 0T kσi (s)k2 ds
0
< +∞.
E(LT ) = 1 et pour tout i = 1, · · · , d, EP LT e
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Then the probability Q defined by dQ|FT = LT dP|FT is an martingale measure equivalent to
P, that is the prices after change of numéraire (S̃ti ) are Q martingales for the filtration (Ft ).
Remark 4.16 The assumptions concerning the convergence of the integrals in condition
(ii) are full-filed as soon as the processes λ and σ i are bounded, or more generally if
h 1 RT
i
kλ(s)k2 +kσi (s)k2 )ds
(
0
2
EP e
< +∞.
Indeed, in that case the local martingale is an exponential martingales.
Proof. Apply the Girsanov theorem. The probability
Q is equivalent to P and the process
j
j Rt j
defined for j = 1, · · · , k and t ∈ [0, T ] by Wt = Bt + 0 λs ds is a Q-martingale. Furthermore,
under Q,
dS̃ti = S̃ti σ i (t)dBt + S̃ti [bi (t) − rt ]dt = S̃ti σ i (t)dWt .
Finally, the second integrability
that the Novikov is satisfied
under Q and
R condition shows
R
t i
1 t
i
2
hence that the process exp 0 σs dWs − 2 0 kσ (s)k ds , t ∈ [0, T ] is a Q-martingale.
The explicit formulation of S̃ti (under Q) in terms of W proved in Exercise 2.4 then concludes
the proof.
2
4.2.2 Complete market
In this section, we suppose that the assumptions of Theorem 4.15 are satisfied. We let
Q denote the probability with density LT with respect to P, such that under Q the processes
(S̃ti , 0 ≤ t ≤ T ) are (Ft , 0 ≤ t ≤ T )-martingales.
Let ζ be a FT -measurable random variable which is Q integrable after the change of
numéraire. We want to prove that the asset with terminal value ζ is replicable. This requires
an assumption on the « rank » of the diffusion matrix σ. Recall that for any t, the diffusion
matrix σ(t) is associated with a linear map from Rk to Rd with adjoint map σ(t)∗ : Rd → Rk
whose matrix in the canonical basis is the transposed σ(t)∗ of the diffusion matrix σ(t). The
linear map σ(t)∗ is onto if and only if the rank of this matrix (which is also that of the
matrix σ(t)) is equal to k.
Theorem 4.17 Let Q be the equivalent martingale measure from Theorem 4.15. The market
is Q-complete if and only if for every t ∈ [0, T ], the rank of σ(t) is a.s. equal to k.
Proof. Let ζ be a FT -measurable random variable such that Sζ0 ∈ L1 (Q). We want to find a
T
self-financing Q-admissible strategy Θ such that VTΘ = ζ. In the previous section, we have
proved that if such a strategy exists, then under Q for all t ∈ [0, T ], using the Q-martingale
and self-financing property :
ṼtΘ
= EQ
ζ Ft
ST0
ζ Ft is a Q
ST0
(FtW ) is included
The process Mt = EQ
=
V0Θ
+
d Z
X
i=1
0
t
θsi S̃si σsi dWs .
(FtW )-martingale. Since the process λ is not determi-
nistic, the filtration
in (FtB ), but does not coincide with it.
Using Lemma 3.1 we deduce that the product (Lt Mt ) is a (FtB , 0 ≤ t ≤ T ) local martingale under P. The martingale representation theorem implies the existence of a unique
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process in H ∈ H2loc (for the filtration (FtB )) such that under P, Mt Lt = M0 +
that is
sous Q , d(Mt Lt ) = Ht dWt − Ht λt dt.
R
R
t
1 t
2
Since L−1
=
exp
λ
dW
−
kλ
k
ds
,
s
s
t
0 s
2 0
under Q ,
Rt
0
Hs dBs ,
−1
dL−1
t = λt Lt dWt .
Then Itô’s formula implies that under Q :
dMt = d[(ML)t L−1
t ]
=
Ht L−1
t (dWt
− λt dt) +
Mt Lt λt L−1
t dWt
Ht
dWt .
+ dhML, L it = Mt λt +
Lt
−1
t
Set Kt = Mt λt + H
; this yields a progressively measurable process K such that under Q,
Lt
Rt
Mt = M0 + 0 Ks dWs . Identifying the decomposition of Mt as an Itô process, we now look
P
for θti , i = 1, · · · , d such that for any t ∈ [0, T ] and any j = 1, · · · , k, Ktj = di=1 θti S̃ti σji (t).
In matrix form, this means that we look for (θi ) such that

 

 

θt1 S̃t1
σ11 (t) · · · σ1d (t)
θt1 S̃t1
Kt1

 
..
..   ..  =  ..  .
σ(t)∗  ...  =  ...
.
.  .   . 
1
d
d d
σk (t) · · · σk (t)
Ktk
θt S̃t
θtd S̃td
This equation must be solved for all right hand sides K (which are integrable and measurable), which implies that rank of the matrix σ(t)∗ must be equal to k.
Conversely, if σ(t) has rank k, then σ(t)∗ is onto and the equation has a solution θti ,
i = 1, · · · , d. In order to satisfy
Pthe self-financing condition (after the change of numéraire),
we deduce that θt0 = Mt − di=1 θti S̃ti . Hence the strategy Θ = (θi , 0 ≤ i ≤ d) is selffinancing, with value ṼtΘ equal to Mt at time t. Since (ṼtΘ , 0 ≤ t ≤ T ) is a (FtB )-martingale,
the strategy Θ is admissible and since VTΘ = MT ST0 = EQ (ζ|FT ) = ζ, it replicates ζ.
2
Notice that the arbitrage-free assumption and the existence of a risk-premium vector λ
imply the existence of an equivalent martingale measure Q (thanks to the Girsanov theorem)
and the fact that the market is Q-complete under rank conditions on σ(t). Theorem 4.13
then implies that the martingale measure is unique.
In order to have the rank of σ(t) equal to k, one needs to have d ≥ k, that is that
there are enough assets to replicate the k sources of randomness B j , 1 ≤ j ≤ k. Finally,
note that if k = d and if for almost every (t, ω) ∈ [0, T ] × Ω the matrix σ(t) is invertible,
then λt = σ(t)−1 [b(t) − rt 1̄]. Hence, if the matrix σ(t, ω) is a.s. bounded and if σ(t)σ(t)∗
is strictly elliptic (that is there exist constants 0 < m < M such that for every y ∈ Rk ,
mkyk2 ≤ y ∗ σ(t)σ ∗ (t)y ≤ Mkyk2 ) and if the coefficients bi (t) are a.e. bounded, then the
conditions of Theorem 4.15 are satisfied.
As soon as the market is complete and arbitrage-free, the price of an contingent claim is
determined in a unique way. This claim is redundant and its price does not depend on the
way investors react to risk.
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Extended Black & Sholes model
4.2.3 Computing the hedging portfolio in the Black & Sholes model
Recall that in this model one has a non-risky asset St0 = exp(rt) with constant
rate r > 0
“
”
σB + b− σ
2
t
2
and a risky asset St which is a geometric Brownian motion St = S0 e t
, solution to
the linear SDE dSt = St (σdBt + bdt) where B is a one-dimensional Brownian motion and
σ > 0. Let (Ft ) the natural filtration generated by B. Collecting the above computations
and the general results on financial markets, we have the following properties.
After a change of numéraire, if S̃t = St e−rt , then dS̃
(b − r)dt. Let λ = b−r
t = σ S̃t + S̃t σ
2
and let Q denote the probability with density LT = exp −λBT − λ2 with respect to P on
FT . Then the process (Wt = Bt + λt, 0 ≤ t ≤ T ) is a (Ft , 0 ≤ t ≤ T )-martingale under Q
such that dS̃t = σ S̃t dWt . Hence the probability Q is an equivalent martingale
measure,
and
σ2
under Q, the process S̃ is the exponential martingale S̃t = S0 exp σWt − 2 . Hence the
non-negative wealth strategies are arbitrage-free. Since σ > 0, the market is complete and
the martingale measure Q is unique. Furthermore, given any ζ ∈ FT such that Z ∈ L1 (Q),
after the change of numéraire the price at time t of the claim which pays the amount ζ at
terminal time T is e−r(T −t) EQ (ζ|Ft). The probability Q is the risk-neutral probability.
One can find a self-financing portfolio made with the basic assets (the non-risky asset
0
S and the risky asset S) which has the same value as the claim at any time t ∈ [0, T ]. The
agent which is selling the claim should use this portfolio which is called the hedging strategy
of the claim. At terminal time, the value of this portfolio is exactly the amount ζ that the
seller is committed to pay.
Theoretically, the portfolio is adjusted continuously, independently of the value of the
underlying risky asset S. The transactions are made continuously and are cost-free ; no
money is added or withdrawn. The amount of risky asset in the hedging strategy is called
the Delta.
In fact, the transactions are made in a discrete way and the transaction costs impose a
limit to the number of adjustments made (called « hedges » ). Thus the seller takes a risk.
The more hedges he does, the closer his portfolio is close to the derivative, but the more
transaction costs he has to pay.
How to hedge to pay the amount h(ST ) at time T
This derivative only depends on the terminal value (as a call or a put), but not on the
entire trajectory (St , t ∈ [0, T ]).
If h(ST ) is Q-integrable, the claim is replicable by a Q-admissible strategy. If Wt = Bt +λt ,
then (Wt , t ∈ [0, T ]) is a Q Brownian motion and the price of the derivative at time t is
σ2
−r(T −t)
EQ e
h(ST )|Ft with ST = St exp σ(WT − Wt ) + r −
(T − t) .
2
The random variable St is Ft -measurable and WT − Wt is independent of Ft . The Markov
property proves that the price can be written V (t, St ) where
”
“
2
σY + r− σ2 (T −t)
−r(T −t)
où Y ∼ N (0, T − t).
V (t, x) = EQ e
h xe
In section 2.6.2 we have proved that the function V (t, x) is solution of the Feynman-Kac
formula
∂V
(t, x) + Lt V (t, x) = rV (t, x), ∀(t, x) ∈ [0, T ] × R
∂t
,
V (T, x) = h(x)
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4
(t, x) +
where Lt f (t, x) = rx ∂f
∂x
σ2 2 ∂ 2 f
x ∂x2 (t,x).
2
−rT
The price at time 0 is V (0, S0 ) = EQ e
Applications to finance
“
” 2
σN (0,T )+ r− σ2 T
h S0 e
.
e finally have to compute the hedging portfolio, that is the amount θt0 of non-risky asset
and ∆t = θt1 of risky asset at time t the self-financing strategy Θ replicating the derivative.
This portfolio is entirely determined by the price of the derivative at time 0 (the initial value
of the portfolio). and the « ∆ », that is the process (∆t ). In section 2.6.2 we have seen that
the decomposition of an Itô process and the self-financing condition imply that
∆t =
∂V
(t, St ).
∂x
(4.4)
The parabolic PDE satisfied by V (t, x) does not depend on b. The delta ∆t measures the
sensitivity of the price of the derivatives to the variations of the price of the underlying risky
asset St . It is the part of the portfolio invested on the risky asset. This hedging strategy is
adjusted in discrete time. If the ∆t varies a lot, one has to change often the composition of
the portfolio and the variations of the delta caused by the value of the underlying asset is
measured by the « gamma » defined by
Γt =
∂2V
(t, St ).
∂x2
(4.5)
Computation of the delta Suppose that h ∈ C 2 has polynomial growth as well as its first
and second derivatives. We only make explicit computations at time 0. The formulas at any
time are similar, replacing
S0 by St .
T by T −σ2t and
Let f (x, ω) = e−rT h xeσWT +(r− 2 )T . Then one has V (0, x) = EQ (f (x, ω)). To find ∆0 ,
one should exchange the derivation with respect to x and the expectation with respect to
Q. Using classical results in measure theory, this is possible provided for Q-almost every ω,
variable
g ∈ L1 (Q) such
the map x → f (x, ω) is of class C 1 , and if there exists a random
∂f
that for every x0 ad every x in a neighborhood of de x0 , ∂x (x, ω) ≤ g(ω) for Q almost
every ω.
Since h′ has polynomial growth, if x remains bounded one has
h
i 2
2
σWT + r− σ2 T
′
σWT − σ2 T ′
h xe
fx (x, ω) = e
for some integrable random variable independent of x. This yields
σ2
∂V
(t, S0 ) = EQ eσWT − 2 T h′ (ST ) .
∆0 =
∂x
A similar computation gives
∆t =
σ2
∂V
(t, St ) = EQ eσWT −t − 2 (T −t) h′ (ST −t ) .
∂x
If the sign of h′ is constant, that is if h is monotone, the sign of ∆0 is that of h′ . Changing
again probability gives a slightly simpler expression of the delta.
2
Let Q̃ denote the probability with density exp(σWT − σ2 T ) with respect to Q. Then the
process defined by W̃t = Wt − σt for 0 ≤ t ≤ T is a (FtB , 0 ≤ t ≤ T )-Brownian motion under
Q̃. Furthermore,
σ2
∆0 = EQ̃ h′ S0 eσW̃T +(r+ 2 )T .
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Computation of the gamma A similar argument allows to derive once more under the
expected value, which yields
2
∂2V
σWT −t − σ2 (T −t) ′′
Γ0 =
(t,
S
)
=
E
e
h
(S
)
.
0
Q
T −t
∂x2
σ2
The new change of probability implies Γ0 = EQ̃ h′′ S0 eσW̃T +(r+ 2 )T .
Hence, if h is convex, the is non-negative.
Price of a call with maturity T and strike K
We apply the previous results for h(x) = (x−K)+ . Under the probability Q, S̃ is solution
2
to the SDE dS̃t = σ S̃t dWt + r − σ2 dt. The computation of the value of the call has been
made in section 2.6.2. Equation (2.39) shows that the value of the call at time t is given by
Ra
x2
(2.39). For any x > 0 equation (2.40) shows that if F (a) = √12π −∞ e− 2 dx,
C(0, x) = EQ e−rT (S̃T − K)+ = S0 F (d1 ) − Ke−rT F (d2 ),
where
S0
σ2
1
ln
+ r+
T
d1 = √
K
2
σ T
√
1
S0
σ2
ln
d2 = d1 − σ T = √
+ r−
T .
K
2
σ T
Replacing S0 by St and T by T − t, one can check as an exercise that equation (2.40)
can be rewritten as follows :
C(0, x) = St F (d1 (t, St )) − Ke−r(T −t) F (d2 ),
where
y
1
σ2
√
ln
d1 (t, y) =
+ r+
(T − t) ,
K
2
σ T −t
√
1
σ2
y
+ r−
(T − t) .
d2 (t, y) = d1 (t, y) − σ T − t = √
ln
K
2
σ T
The function h(y) = (y − K)+ has a derivative at any point y with y 6= K and the set
σ2
of ω such that xeσWT +(r− 2 )T (ω) = K is a null set. Furthermore, dy-almost everywhere
h′ (y) = 1]K,+∞[(y). The above argument (derivation under the Q expected value) shows
that using the probability Q̃, one has
h
i
σ2
∆0 = EQ̃ 1]K,+∞[(S0 eσW̃T +(r+ 2 )T .
Since W̃ is a Q̃ Brownian motion, we deduce
σ2
S0
+ r + )T
= F (d1 (0, S0 )).
∆0 = Q̃ σ W̃T ≤ ln
K
2
A similiar computation proves that ∆t = F (d1 (t, St )), that is gives the composition of the
hedging portfolio :
Ct = ∆t St + θt0 ert , où θt0 = −Ke−rT F (d2(t, St )) ≤ 0.
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4.2.4 Volatility
The above formulas of the price and the delta depend on a parameter of the underlying
asset S which is not directly observable, namely the volatility σ. Indeed, the drift coefficient
b does appear in the equations giving Ct and ∆t .
One usually has two ways to estimate σ.
Historical Volatility One wants to estimate σ from the observations of the past values of
the underlying asset. Thus, one studies (Sih , 0 ≤ i ≤ n) at times which are multiples
of a fixed
Sih
time unit (e.g., one day). For every i = 1, · · · , n, the random variables Xi = ln S(i−1)h
are
i
h
2
independent with the same normal distribution N b − σ2 h, σ 2 h . The classical estimator
of the
variance of a n-sample whose mean is unknown is an unbiased estimator of σ 2 (since
P
n
1
2
2
i=1 (Xi − X̄n ) has a χn−1 distribution) :
σ2
n
1 X
(Xi − X̄n )2
n − 1 i=1
n
where
1X
X̄n =
Xi .
n i=1
The size of the sample is often the life time of the derivative until T .
Implied Volatility
One sees that the price of the call is a strictly increasing function of σ. Observing the
prices of puts and calls with different maturities, one finds σ by inverting the formula.
In fact, the various derivatives give different volatilities. The way it depends on K can
be summarized by some « smile » , that is the options for « small » or for « large » values of
K are more expensive and have a larger implied volatility. Two things can explain this fact :
one one hand, the model is ”wrong’ (mainly the fact that the volatility remains constant)
and on the other hand, the derivatives for which the strikes are extreme are not liquid, that
is not very much sold or bought.
Traders rather think in terms of volatility and smile of volatility and only use the Black
& Sholes model to express the link between the price and the implied volatility. The models
which can take care of the volatility smile are those where the volatility σt is random and
depends on time. Considering non-constant but deterministic volatility is not enough.
Several models are used.
The Hull & White model Let B and W be two independent Brownian motions. The
value of the underlying asset St is modeled by the SDE drive by B with diffusion coefficient
σt St and with drift coefficient bt St . The volatility σt is random and its square Vt is solution
of a linear SDE driven by W with diffusion coefficient St Vt and drift coefficient Mt Vt . This
yields the system :
dSt = St (σt dBt + bt dt),
dVt = Vt (st dWt + mt dt), où Vt = σt2 .
In that case, the price of a call can be expressed in terms of that in the Black & Sholes
model.
The Dupire model The volatility is a function σ(t, St ) with 0 < α ≤ σ(t, St ) ≤ β. Then
the SDE has a unique solution, but no explicit formulation of this solution is available.
Jump processes One tries to model the value of a non-continuous underlying asset by
means of processes solutions to SDEs driven by processes different from the Brownian motion, such as Poisson or Lévy processes.
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The Cox-Ingersoll-Ross model
4.3 The Cox-Ingersoll-Ross model
An important class of stochastic models (more recent than the Black & Sholes one) used
in finance is based on Bessel processes.
4.3.1 General Bessel processes
Bessel processes have been introduced in Chapter 2. We will now define more general
Bessel processes.
p
√
√
Let B be a Brownian motion. Since | x − y| ≤ |x − y|, the Yamada-Watanabe
Theorem 2.20 shows that for any δ ≥ 0 and α ≥ 0, the SDE
p
dZt = δ dt + 2 |Zt | dBt , Z0 = α ,
has a unique strong solution. This solution is a square Bessel process of dimension δ, which
we will denote by BESQδ . In particular, if α = 0 et δ = 0, then Z ≡ 0 is the unique solution.
Using the comparison theorem, we deduce that if 0 ≤ δ ≤ δ ′ and if ρ and ρ′ are square
Bessel processes of dimension δ and δ ′ starting from the same point, then 0 ≤ ρt ≤ ρ′t a.s.
When δ > 2, the square Bessel process BESQδ starting from α > 0 never reaches 0. If
0 < δ < 2, the process ρ reaches 0 in finite time. If δ = 0 the process stays in 0 when it
reaches it.
Definition 4.18 (BESQδ ) For any δ ≥ 0 and α ≥ 0, the unique strong solution to the SDE
ρt = α + δt + 2
Z
t
√
ρs dBs
0
is a square Bessel process with dimension δ, starting from α and is denoted BESQδ .
Definition 4.19 (BESδ ) Let ρ be a BESQδ starting from α.The process R =
√
process of dimension δ, starting from a = α and is denoted BESδ .
√
ρ is a Bessel
Definition 4.20 The real number ν = (δ/2) − 1 (that is δ = 2(ν + 1)) is the index of the
Bessel process and a Bessel process with index ν is denoted by BES(ν) .
Let δ > 1 ; then a BESδ is solution of the SDE
δ−1
Rt = α + Bt +
2
Z
0
t
1
ds .
Rs
(4.6)
The modified Bessel functions are the solutions Iν and Kν of the following equations
x2 u′′ (x) + xu′ (x) − (x2 + ν 2 )u(x) = 0
and are equal to :
Iν (z) =
∞
z ν X
2
n=0
z 2n
22n n! Γ(ν + n + 1)
π(I−ν (z) − Iν (z))
.
Kν (z) =
2 sin πz
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Applications to finance
(ν)
of a BESQ(ν) are
√
xy
1 y ν/2
x+y
(ν)
Iν (
qt (x, y) =
exp −
)
2t x
2t
t
The transition probabilities qt
(4.7)
(ν)
and the transition function of a Bessel process with index ν is the function pt
y y ν
x2 + y 2 xy (ν)
pt (x, y) =
,
)Iν
exp(−
t x
2t
t
defined by :
(4.8)
Bessel processes are used in the following Cox-Ingersoll-Ross model.
4.3.2 The Cox-Ingersoll-Ross model
To modem rates, Cox-Ingersoll-Ross have introduced the following equation
√
drt = k(θ − rt )dt + σ rt dBt
(4.9)
Using the Yamada-Watanabe Theorem (Theorem 2.20) we deduce that (4.9) has a unique
solution. This solution is a non-negative process r kθ ≥ 0, but the solution can not be given
explicitly. Let r x denote the process solving (4.9) with initial condition r0x = x.
Using the change of time A(t) = σ 2 t/4, the study of (4.9) can be reduced to the case
σ = 2. Indeed, if Zt = rσ2 t/4 , then
p
dZt = k ′ (θ − Zt ) dt + 2 Zt dBt ,
with k ′ = kσ 2 /4 and where B is a Brownian motion.
The CIR process solution of equation (4.9) is a time-change of a BESQ. Indeed,
2
σ kt
−kt
(e − 1 )
rt = e ρ
4k
where (ρ(s), s ≥ 0) is a BESQδ (α), and δ = 4kθ
.
σ2
One can prove that if T0x := inf{t ≥ 0 : rtx = 0} et 2kθ ≥ σ 2 then P (T0x = ∞) = 1. If
0 ≤ 2kθ < σ 2 and k > 0 then P (T0x < ∞) = 1 and if k < 0 then P (T0x < ∞) ∈]0, 1[. (These
results are proved by using the comparison theorem.)
One can give explicit expressions of the mean of the random variable rt using the equality
Z t
E(rt ) = r0 + k(θt −
E(rs )ds),
0
and supposing that the stochastic integral is a martingale, which is indeed the case. It is
not more difficult to identify the conditional expectation thanks to the Markov property.
Theorem 4.21 Let r be the stochastic process defined by
√
drt = k(θ − rt )dt + σ rt dBt .
The the conditional expectation and the conditional variance are given by the following equations :
E(rt |Fs ) = rs e−k(t−s) + θ(1 − e−k(t−s) ),
Var(rt |Fs ) = rs
σ 2 (e−k(t−s) − e−2k(t−s) ) θσ 2 (1 − e−k(t−s) )2
+
.
k
2k
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The Cox-Ingersoll-Ross model
Proof. The definition yields for any s ≤ t
Z t
Z t
√
rt = rs + k (θ − ru )du + σ
ru dBu ,
s
s
and applying Itô’s formula, we deduce
Z t
Z t
Z t
2
2
3/2
2
rt = rs + 2k (θ − ru )ru du + 2σ (ru ) dBu + σ
ru du
s
s
s
Z t
Z t
Z t
2
2
2
= rs + (2kθ + σ )
ru du − 2k
ru du + 2σ
(ru )3/2 dBu .
s
s
s
Let us assume that the stochastic integrals appearing in the above identities have zero mean ;
this yields for s = 0
Z
t
E(rt ) = r0 + k θt −
et
E(rt2 )
=
r02
2
+ (2kθ + σ )
Z
0
E(ru )du ,
0
t
E(ru )du − 2k
Z
Set Φ(t) = E(rt ). Solving the equation Φ(t) = r0 + k(θt −
Φ′ (t) = k(θ − Φ(t)) and Φ(0) = r0 , we obtain
t
E(ru2 )du.
0
Rt
0
Φ(u)du), that is the ODE
Φ(t) = E[rt ] = θ + (r0 − θ)e−kt .
Similarly, set Ψ(t) = E(rt2 ) and solve the equation Ψ′ (t) = (2kθ + σ 2 )Φ(t) − 2kΨ(t) ; this
yields
θ
σ2
−kt
−kt
−kt
(1 − e ) r0 e + (1 − e ) .
Var [rt ] =
k
2
The conditional expectation and mean are obtained in a similar way, using the Markov
property :
E(rt |Fs ) = θ + (rs − θ)e−k(t−s) = rs e−k(t−s) + θ(1 − e−k(t−s) ),
σ 2 (e−k(t−s) − e−2k(t−s) ) θσ 2 (1 − e−k(t−s) )2
+
.
k
2k
RT
− t ru du Let us compute E e
Ft .
Var(rt |Fs ) = rs
2
4.3.3 Price of a zero-coupon
Proposition 4.22 Let
Then
√
drt = a(b − rt )dt + σ rt dBt .
−
E e
with
RT
t
ru du
Ft = G(t, rt ),
G(t, x) = Φ(T − t) exp[−xΨ(T − t)],
2ab
s
σ2
2γe(γ+a) 2
2(eγs − 1)
,
Φ(s)
=
, γ 2 = a2 + 2σ 2 .
Ψ(s) =
(γ + a)(eγs − 1) + 2γ
(γ + a)(eγs − 1) + 2γ
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Applications to finance
Proof. Let (rsx,t , s ≥ t) denote the solution of the SDE
p
drsx,t = a(b − rsx,t)ds + σ rsx,tdBs , rtx,t = x
and for all s ≥ t set
Rst
Z s
x,t
= exp −
ru du .
t
The Markov property implies the existence of G such that
Z s
x,t
exp −
ru duFt = G(t, rt ).
t
Let us assume that G if of class C 1,2 . Itô’s formula applied to G(s, rsx,t)Rst , which is a
martingale, shows that
G(T, rTx,t)RTt = G(t, x) + MT − Mt
Z T 1 2 x,t ∂ 2 G
∂G
x,t
x,t ∂G
x,t
x,t
t
x,t
x,t
(s, rs ) + a(b − rs )
(s, rs ) + σ rs
(s, rs ) ds,
+
Rs −rs G(s, rs ) +
∂t
∂x
2
∂x2
t
where Mt is a stochastic integral. Computations similar to those proving the Feynman-Kac
formula, choosing G such that
−yG(s, y) +
∂G
1
∂2G
∂G
(s, y) + a(b − y)
(s, y) + σ 2 y 2 (s, y) = 0
∂s
∂y
2
∂y
(4.10)
and G(T, y) = 1 for all y, yield
RTt = G(t, x) + MT − Mt ,
R
T
where M is a martingale. In particular, when t = 0 we obtain E exp − 0 rs ds
=
E(RT ) = G(0, x). Using this formula between times t and T , we deduce
R T x,t E e− t ru du = G(t, x).
We still should solve the PDE (4.10). A lengthy computation process that
G(t, x) = Φ(T − t) exp[−xΨ(T − t)],
with
γs
2(e −1)
Ψ(s) = (γ+a)(e
γs −1)+2γ ,
γ 2 = a2 + 2σ 2 .
Φ(s) =
s
2γe(γ+a) 2
(γ+a)(eγs −1)+2γ
2ab2
σ
,
2
Let P (t, T ) denote the price of the associated zero-coupon,
P (t, T ) = Φ(T − t) exp[−rt Ψ(T − t)],
we deduce that
P (t, T ) = B(t, T ) (rt dt + σ(T − t, rt )dBt ) ,
√
with σ(u, r) = σΨ(u) r.
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Exercises
4.4 Exercises
Exercise 4.1 Let σ and b be constants, r > 0, x > 0, and (St ) denote the geometric
Brownian motion solution to the SDE
dSt = St σdBt + bdt , S0 = x.
1. Write St as an exponential.
1 2
2. Set θ = − b−r
, Lt = eθBt − 2 θ t and Q the probability defined on Ft by dQ = Lt dP.
σ
Construct a un Brownian motion (Wt ) under Q. Which SDE does (St ) solve under Q ?
3. Let P̃ denote the probability defined on Ft by dP̃ = Zt dQ where Zt = eσWt −
that
dSt = St σdB̃t + (r + σ 2 )dt ,
where B̃ is a P̃-Brownian motion.
rt
St
,t
Pt
σ2
t
2
. Prove
4. Let P0 > 0 and for every t ≥ 0, Pt = P0 e . Show that
≥ 0 is a Q-martingale.
Prove that PStt , t ≥ 0 is a P̃-martingale.
R
rt
t
5. Let A and λ be real numbers, Ft = e−λt 0 Su du + xA and Ψt = FtSet . Write the
SDE satisfied by Ψt using the P̃-Brownian motion B̃.
Exercise 4.2 Stochastic Volatility Let B 1 and B 2 be independent Brownian motions,
Ft = σ(Bs1 , Bs2 , s ≤ t) denote the filtration generated by B 1 and B 2 . Let µ and η be
(deterministic) bounded functions from [0, +∞[ in R, σ and γ be (deterministic) bounded
functions from R in [m, +∞[ with m > 0. Let S denote the solution to the SDE
dSt = St σ(Yt )dBt1 + µ(t)dt , S0 = x ∈ R,
where (Yt ) solves the SDE
dYt = γ(Yt )dBt2 + η(t)dt , Y0 = 1.
1. Let θ denote a bounded, (Ft ) adapted, real-valued process and (Zt ) the solution to
the SDE
dZt = Zt θt dBt1 , Z0 = 1.
Write explicitly Zt as an exponential.
2. Let λ and ν be bounded, (Ft )-adapted processes and let (Lt ) be the process defined
by
Z t
Z
Z
Z t
1 t 2
1 t 2
1
2
Lt = exp
λs dBs −
λ ds +
ν ds .
(4.11)
νs dBs −
2 0 s
2 0 s
0
0
Write the SDE satisfied by (Lt ). Let Q denote the probability defined by dQ = Lt dP
on Ft .
Rt
3. Let B̃t1 = Bt1 − 0 λs ds and (Z̃t ) be the solution to the SDE dZ̃t = Z̃t αdB̃t1 et Z̃0 = 1,
where α is a constant. Prove that (Lt Z̃t ) is a (Ft )-martingale under P. Construct a
process (B̃t2 ) such that (B̃t1 , B̃t2 ) is a 2-dimensional Brownian motion.
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Applications to finance
4. Describe the set of pairs (λ, ν) such that under Q, the process (S̃t = St e−rt , t ≥ 0) is
a martingale. The corresponding set of probabilities will be denoted Q.
5. Is the financial market complete ? Let X be a replicable asset in the following sense :
there exists a (Ft )-adapted process (Vt ) such for some bounded (Ft )-adapted process
(φt ) it holds :
dVt = rVt dt + φt dSt − rSt dt , et VT = X.
(a) Prove that for any Q ∈ Q, under Q the process (Vt e−rt ) is a (Ft )-martingale.
(b) Suppose that Vt = v(t, St , Yt ). Prove that v solves a PDE which should be written
explicitly.
Exercise 4.3 Power Options Let r, δ, σ be positive constants, x > 0. We describe the
dynamics of an asset which pays dividends with the constant rate δ while the spot rate
under the neutral-risk probability is given by :
dSt = St σdBt + (r − δ)dt , S0 = x.
1. We want to compute the
of a contingent
claim based on S which pays dividends,
value −r(T
−t) that is to compute EP h(ST )e
Ft . Using the technique used to establish the
formula in the classical case, what is the value of this claim when h(x) = (xα − K)+
where α > 0 ?
2. Suppose that δ = r and let Q denote the probability on Ft with density Sxt with respect
2
to P. Why is Q a probability ? Set Zt = xSt . Describe the dynamics of (Zt , t ≥ 0) under
Q. Prove that for every bounded Borel function f
2 x
1
= EP (f (ST )).
EP ST f
x
ST
3. Let us come back to the general case. Prove that (Sta , t ≥ 0) is a martingale for a value
of a which should be given explicitly. Prove that for every bounded Borel function f ,
2 1
x
a
EP [f (ST )] = α EP ST f
.
x
ST
4. Suppose that h(x) = xβ (x − K)+ . Prove that h(ST ) is the difference of two pay-offs
for European calls based on the assets S β+1 and S β and strikes which will be given
explicitly.
Acknowledgements There lectures replace those given by Isabelle Nagot (Calcul Stochastique et Finance) in the former DEA MME University of Paris 1. It is a continuation of the
lectures Calcul Stochastique and then Stochastic Calculus 1 from the University Paris 1 given by Bernard de Meyer then by Ciprian Tudor respectively. I thank these three colleagues
from Paris 1 for giving me their notes.
I also want to express my gratitude to Monique Jeanblanc and Thomas Simon, who
gave me the most recent version of the notes of their lectures on stochastic calculus and
applications in finance they produced in the Masters of the University of Evry.
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Références
Références
[1] Comets, F., Meyre, T., Calcul stochastique et modèles de diffusions, Dunod 2006.
[2] De Meyer, B., Calcul Stochastique, Polycopié du cours de l’ex DEA MME de l’Université Paris 1, année 2004-2005.
[3] Friedman, A., Stochastic Differential Equations and Applications, Volume 1, Academic
Press, 1975.
[4] Jeanblanc ; M., Cours de calcul stochastique, DESS IM Evry, Septembre 2005,
http ://www.maths.univ-evry.fr/pages perso/jeanblanc/
[5] Jeanblanc, M., Simon, T., Eléments de calcul stochastique, IRBID, Septembre 2005.
[6] Jeanblanc, M., Yor, M. et Chesney, M., Mathematical Methods for financial Markets,
Springer Verlag, à paraı̂tre.
[7] Karatzas, I., Shreve, S.E., Brownian motion and Stochastic Calculus, Springer Verlag,
1991.
[8] Lamberton, D., Lapeyre., B, Introduction au calcul stochastique appliqué à la finance.,
Ellipses, Paris, 1991.
[9] Malliavin P., Stochastic Analysis, Springer, Berlin, 1997.
[10] Nagot, I., Calcul Stochastique et Finance, Temps continu, Polycopié du cours de l’ex
DEA MME de l’Université Paris 1, année 2004-2005.
[11] Revuz, A., Yor, M., Continuous Martingales and Brownian Motion, Springer, Berlin,
3th edition 1999.
[12] Rogers L. C. G. et Williams, D., Diffusions, Markov Processes, and Martingales. Volume
1 : Foundations, Wiley, Chichester, 1994.
[13] Rogers L. C. G., Williams, D., Diffusions, Markov Processes, and Martingales. Volume
2 : Itô Calculus, Wiley, Chichester, 1987.
[14] Tudor, C., Calcul Stochastique 1, Cours polycopié du Master M2 MMEF de l’Université
Paris 1, année 2005-2006, http ://pagesperso.aol.fr/cipriantudor/
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