Download Stochastic Differential Equations (SDEs)

Stochastic Differential Equations (SDEs)
Francesco Russo
Université Paris 13, Institut Galilée, Mathématiques
99, av. JB Clément, F-99430 Villetaneuse, France
October 4th 2005
1
Introduction
This paper will appear in Encyclopedia of Mathematical Physics, see
http://www1.elsevier.com/homepage/about/mrwd/emp/menu.htm.
and it constitutes a survey paper on the title subject.
Stochastic differential equations (SDEs) appear today as a modeling
tool in several sciences as telecommunications, economics, finance, biology and quantum field theory.
A stochastic differential equation is essentially a classical differential
equation which is perturbed by a random noise. When nothing else is
specified, SDE means in fact ordinary stochastic differential equation;
in that case it corresponds to the perturbation of an ordinary differential equation. Stochastic partial differential equations (SPDEs) are
obtained as random perturbation of partial differential equations.
One of the most important difference between deterministic and stochastic ordinary differentials equation is described by the so called Peano
type phenomenon. A classical differential equation with continuous and
linear growth coefficients admits global existence but not uniqueness as
1
Stochastic Differential Equations.
2
classical calculus text books illustrate studying equations of the type
!
dX
(t) = X(t),
dt
X(0) = 0.
However if one perturbes the right member of the equality with an
additive Gaussian white noise (ξt ) (even with very small intensity),
then the problem becomes well-stated. A similar phenomenon happens
with linear PDEs of evolution type perturbed with a space-time white
noise.
Stochastic differential equations constitute a very huge subject and an
incredible amount of relevant contributions were brought. We try to
orientate the reader about the main axes trying to indicate references
to the different subfields. We will prefer to refer to monographies when
available, instead of articles.
2
Motivation and preliminaries
In the whole paper T will be a strictly positive real number. Let us
consider continuous functions b : R+ × Rd → Rd , a : R+ × Rd×m → Rd
and x0 ∈ Rd . We consider a differential problem of the following type.
"
dXt
= b(t, Xt )
dt
(2.1)
X0 =
x0 .
Let (Ω, F , P) be a complete probability space. Suppose that previous
equation is perturbed by a random noise (ξt )t≥0 . Because of modeling
reasons it could be reasonnable to suppose (ξt )t≥0 verifying the following
properties.
1) It is a family of independent random variables.
2) (ξt )t≥0 is stationary i.e. for any positive integer n, positive reals
h, t0 , t1 , . . . , tn the law of (ξt0 +h , . . . , ξtn +h ) does not depend on h.
Stochastic Differential Equations.
3
More precisely we perturb previous equation (2.1) as follows:
"
dXt
= b(t, Xt ) + a(t, Xt )ξt
dt
X0 =
x0 .
(2.2)
We suppose for a moment that d = m = 1. In reality no reasonnable
real valued process (ξt )t≥0 fulfilling previous assumptions exists. In
particular, if process (ξt ) exists (resp. exists and each ξt is a square
integrable random variable) then the process cannot have continuous
paths (resp. it cannot be measurable with respect to Ω×R+ ). However,
#t
suppose that such a process exists; we set Bt = 0 ξs ds. In that case,
previous properties 1) and 2) can be translated into the following on
(Bt ).
(i) It has independent increments, that means that for any t0 , . . . , tn , h ≥
0, Bt1 +h − Bt0 +h , . . . , Btn +h − Btn−1 +h are independent random
variables.
(ii) It has stationary increments, that means that for any t0 , . . . , tn , h ≥
0, the law of (Bt1 +h −Bt0 +h , . . . , Btn +h −Btn−1 +h ) does not depend
on h.
On the other hand it is natural to require that
a) B0 = 0 a.s.,
b) it is a continuous process, i.e. it has continuous paths a.s.
Equation (2.2) should be rewritten in some integral form
$ t
$ t
b(s, Xs )ds +
a(s, Xs )dBs .
Xt = X0 +
0
(2.3)
0
Clearly the paths of process (Bt ) cannot be differentiable so one has to
#t
give meaning to integral 0 a(s, Xs )dBs . This will be intended in the
Itô sense, see considerations below.
Stochastic Differential Equations.
4
An important result of probability theory says that a stochastic process
(Bt ) fulfilling properties (i), (ii) and a) b) is essentially a Brownian
motion. More precisely, there are real constants b, σ such that Bt =
bt + σWt where (Wt ) is a classical Brownian motion defined below.
Definition 2.1
(i) A (continuous) stochastic process (Wt ) is called
classical Brownian motion if W0 = 0 a.s., it has independent
increments and the law of Wt − Ws is a Gaussian N(0, t − s)
random variable.
(ii) A m-dimensional Brownian motion is a vector (W 1 , . . . , W m )
of independent classical Brownian motions.
Let (Ft )t≥0 be a filtration fulfilling the usual conditions, see Section
1.1 of [9]. There one can find basic concepts of the theory of stochastic processes as the concept of adapted, progressively measurable
process. An adapted process is also said to be non-anticipating towards
the filtration (Ft ) which represents the state of the information at each
time t. A process (Xt ) is said to be adapted if for any t, Xt is Ft -
measurable. The notion of progressively measurable process is a slight
refinement of the notion of adapted process.
Definition 2.2
(i) A (continuous) (Ft ) adapted process (Wt ) is called
(classical) (Ft )-Brownian motion if W0 = 0, if for any s < t
Wt − Ws is an N(0, t − s) distributed random variable which is
independent of Fs .
(ii) An (Ft )-m-dimensional Brownian motion is a vector (W 1 , . . . , W m )
of (Ft )-classical independent Brownian motions.
From now on we will consider a probability space (Ω, F , P) equipped
with a filtration (Ft )t≥0 fulfilling the usual conditions. From now on all
the considered filtrations will have that property.
Stochastic Differential Equations.
5
Let W = (Wt )t≥0 be an (Ft )t≥0 - m-dimensional classical Brownian motion. In chapter 3 of [9] and chapter 4 of [17] one introduces the notion
of stochastic Itô integral announced before. Let Y = (Y 1 , . . . , Y m ) be a
#T
progressively measurable m- dimensional process such that 0 &Ys &2 ds <
#T
∞, then the Itô integral 0 Ys dWs is well defined. In particular the in#·
definite integral. 0 Ys dWs is an (Ft )-progressively measurable continu#t
ous process. If Y is a Rd×m matrix-valued process, the integral 0 Ys dWs
is componentwise defined and it will be a vector in Rd . The analogous
of differential calculus in the framework of stochastic processes, is Itô
calculus, see again [9] ch. 3 and ch. IV of [17]. Important tools
are the concept of quadratic variation [X] of a stochastic process
when it exists. For instance the quadratic variation [W ]t of a classical
#t
#t
Brownian motion equals t. If Mt = 0 Ys dWs then [M]t = 0 &Ys &2 ds.
One celebrated theorem of P. Lévy states the following: if (Mt ) defines
a continuous (Ft )-local martingale such that [Mt ] ≡ t, then M is an
(Ft )-classical Brownian motion. That theorem is called the Lévy caracterization theorem of Brownian motion. Itô formula constitutes the
natural generalization of fundamental theorem of differential calculus
to the stochastic calculus. Another significant tool is Girsanov theorem; it states essentially the following: suppose that the following so
called Novikov condition is verified:
& &
%
% $ T
1
2
&Yt & dt < ∞.
E exp
2 0
#t
Then the process W̃t = Wt + 0 Ys ds, t ∈ [0, T ] is again a m-dimensional
a new probability( measure Q on
(Ft )- classical Brownian motion under
'#
t
(Ω, FT ) defined by dQ = dP exp 0 Ys dWs − 12 &Ys &2 ds .
Let ξ be an F0 -measurable random variable, for instance ξ ≡ x ∈ Rd .
We are interested in the stochastic differential equation
"
dXt = a(t, Xt )dWt + b(t, Xt )dt
X0
=
ξ.
(2.4)
Stochastic Differential Equations.
6
Definition 2.3 A progressively measurable process (Xt )t∈[0,T ] is said to
be solution of (2.4) if a.s.
$ t
$ t
Xt = Z +
a(t, Xt )dWt +
b(t, Xt )dt,
0
0
∀t ∈ [0, T ].
(2.5)
provided that the right member makes sense. In particular, such a solution is continuous. The function a (resp. b) is called the diffusion
(drift) coefficient of the SDE. a and b may sometimes be allowed to
be random; however this dependence has to be progressively measurable. Clearly, we can define the notion of solution (Xt )t≥0 on the whole
positive real axis.
We remark that those equations are called Itô stochastic differential equations. A solution of previous equation is named diffusion
process.
3
The Lipschitz case
The most natural framework for studying existence and uniqueness
for stochastic differential equations appears when the coefficients are
Lipschitz.
A function γ : [0, T ] × Rm −→ Rd is said to have polynomial growth
(with respect to x uniformly in t), if for some n there is a constant
C > 0 with
sup &γ(t, x)& ≤ C(1 + &x&n )
(3.6)
t∈[0,T ]
The same fonction is said to have linear growth if (3.6) holds with n =
1. A function γ : R+ × Rm → Rd is said to be locally Lipschitz (with
respect to x uniformly in t), if for every t ∈ [0, T ], K > 0, γ|[0,T ]×[−K,K]
is Lipschitz (with respect to x uniformly with respect to t).
Let a : R+ × Rd×m −→ Rd , b : R+ × Rd −→ Rd , be Borel functions,
ξ an Rd -valued random variable F0 -measurable and (Wt )t≥0 be a m-
dimensional (Ft )-Brownian motion.
Stochastic Differential Equations.
7
Classical fixed point theorems allow to establish the following classical
result.
Theorem 3.1 We suppose a and b locally Lipschitz with linear growth.
Let ξ be a square integrable r.v. that is F0 -measurable. Then (2.4) has
a unique solution X. Moreover,
%
&
2
E sup |Xt | < ∞.
t≤T
Remark 3.2
(i) Equation (2.4) can be settled similarly by putting
initial condition x at some time s. In that case the problem is
again well stated. If ξ ≡ x is a deterministic point of Rd than we
will often denote by X s,x the solution of that problem.
(ii) If the coefficients are only locally Lipschitz, the equation may be
solved until a stopping time. If d = 1 it is possible to state necessary and sufficient conditions for non-explosion (Feller test).
(iii) The theorem above admits several generalizations. For instance
the Brownian motion can be replaced by a general semimartingales, (possibly with jumps as Lévy processes).
An important role of diffusion processes is the fact that they provide
probabilistic representation to partial differential equations of parabolic
(and even elliptic) type. We will only mention here the parabolic framework.
We denote A(t, x) = a(t, x)a(t, x)∗ where ∗ means transposition for
matrices. (t, x) → A(t, x) = (Aij (t, x)) is a d×d matrix valued function.
Let us consider also continuous functions k : [0, T ] × Rd → Rd , g :
[0, T ] × Rd → Rd with polynomial growth or non-negative.
Given a solution of (3.1), we can associate its generator (Lt , t ∈ [0, T ])
setting
d
1)
Aij (t, x)∂ij2 f (x) + b(t, x) · ∇f (x).
Lt f (x) =
2 i,j=1
Stochastic Differential Equations.
8
Feynmann-Kac theorem allows to represent probabilistically as follows
Theorem 3.3 Suppose there is a function v : [0, T ] × Rd → Rd con-
tinuous with polynomial growth of class C 1,2 ([0, T [×Rd ) satisfying the
following Cauchy problem
"
(∂t v + Lt )v − kv =
v(T, x)
g
(3.7)
= f (x).
Then
%
$
v(s, x) = E f (XT ) exp(−
T
s
k(θ, Xθ )dθ) −
$
T
g(t, Xt ) exp{−
s
$
s
t
&
k(θ, Xθ )dθ)}dt
for (s, x) ∈ [0, T ] × Rd where X = X s,x . In particular, such a solution
is unique.
Remark 3.4
(i) In order to obtain “classical solutions” of previous
Cauchy problem one needs some conditions. It is the case for
instance when the following ellipticity condition holds on A:
∃c > 0,
∀(t, x) ∈ [0, T ] × Rn
(3.8)
∀(ξ1 , . . . , ξn ) ∈ Rn :
d
)
i,j
Aij (t, x)ξi ξj ≥ c
d
)
i=1
|ξi|2 .
In the degenerate case it is possible to deal with viscosity solutions,
in the sense of P.L. Lions. This theorem establishes an important link between deterministic PDEs and stochastic differential
equations.
(ii) A natural generalization of Feynmann-Kac theorem comes from
the system of forward-backward stochastic differential equations
in the sense of Pardoux and Peng.
(iii) Other types of probabilistic representation do appear in stochastic
control theory through the so-called verification theorems, see for
Stochastic Differential Equations.
9
instance [6, 20]. In that case the (non-linear) Hamilton-JacobiBellmann deterministic equation is represented by a controlled
stochastic differential equation.
(iv) Another bridge between non-linear PDEs and diffusions can be
provided in the framework of interacting particle systems with
chaos propagation, see [7] for a survey on those problems. Among
the most significant non-linear PDEs investigated probabilistically,
we quote the case of porous media equations. For instance, for a
positive integer m, a solution to
1 2 2m+1
(u
)
∂t u = ∂xx
2
(3.9)
can be represented by a (non-linear) diffusion of the type, see [1],
"
um (s, Xs )dWt
dXt =
(3.10)
u(t, ·) = law density of Xt .
4
Different notions of solutions
Let a and b as at the beginning of section 3. Let (Ω, F , P) be a probabil-
ity space, a filtration (Ft )t≥0 fulfilling the usual conditions, an (Ft )t≥0 classical Brownian motion (Wt )t≥0 . Let ξ be an F0 -measurable random
variable. At section 2 we defined the notion of solution of the following
equation:
"
dXt = b(t, Xt )dt + a(t, Xt )dWt
X0
=
ξ.
(4.11)
Previous equation will be denoted by E(a, b) (without initial condition).
However, as we will see, the general concept of solution of an SDE
is more sophisticated and subtle than in the deterministic case. We
distinguish several variants of existence and uniqueness.
Definition 4.1 (Strong existence)
We will say that equation E(a, b) admits strong existence if the following holds. Given any probability space (Ω, F , P), a filtration (Ft )t≥0 ,
Stochastic Differential Equations.
10
an (Ft )t≥0 - Brownian motion (Wt )t≥0 , a F0 -measurable and square in-
tegrable random variable ξ, there is a process (Xt )t≥0 solution to E(a, b)
with X0 = ξ a.s.
Definition 4.2 (Pathwise uniqueness) We will say that equation
E(a, b) admits pathwise uniqueness if the following property is fulfilled.
Let (Ω, F , P) be a probability space, a filtration (Ft )t≥0 , an (Ft )t≥0
Brownian motion (Wt )t≥0 . If two processes X, X̃ are two solutions such
that X0 = X̃0 a.s., then X and X̃ coincide.
Definition 4.3 (Existence in law or weak existence) Let ν be a
probability law on Rd . We will say that E(a, b; ν) admits weak existence
if there is a probability space (Ω, F , P), a filtration (Ft )t≥0 , an (Ft )t≥0 Brownian motion (Wt )t≥0 and a process (Xt )t≥0 solution of E(a, b) with
ν being the law of X0 .
We say that E(a, b) admits weak existence if E(a, b; ν) admits weak
existence for every ν.
Definition 4.4 (Uniqueness in law) Let ν be a probability law on Rd .
We say that E(a, b; ν) has a unique solution in law if the following
holds. We consider an arbitrary probability space (Ω, F , P) and a filtra-
tion (Ft )t≥0 on it; we consider also another probability space (Ω̃, F̃ , P̃)
equipped with another filtration (F̃t )t≥0 ; we consider an (Ft )t≥0 -Brownian
motion (Wt )t≥0 , and an (F̃t )t≥0 - Brownian motion (W̃t )t≥0 ; we suppose
having a process (Xt )t≥0 (resp. a process (X̃t )t≥0 ) solution of E(a, b) on
the first (resp. on the second) probability space such that both the law
of X0 and X̃0 are identical to ν. Then X and X̃ must have the same
law as r.v. with values in E = C(R+ ) (or C[0, T ]).
We say that E(a, b) has a unique solution in law if E(a, b; ν) has a
unique solution in law for every ν.
There are important theorems which establish bridges among the preceding notions. One of the most celebrated is the following.
Stochastic Differential Equations.
11
Proposition 4.5 (Yamada-Watanabe) Consider the equation E(a, b).
i) Pathwise uniqueness implies uniqueness in law.
ii) Weak existence and pathwise uniqueness imply strong existence.
A version can be stated for E(a, b; ν) where ν is a fixed probability law.
Remark 4.6 i) If a and b are locally Lipschitz with linear growth,
Theorem 3.1 implies that E(a, b) admits strong existence and pathwise uniqueness.
ii) If a and b are only locally Lipschitz, then pathwise uniqueness is
fulfilled.
4.1
Existence and uniqueness in law
A way to create weak solutions of E(1, b) when (t, x) → b(t, x) is Borel
with linear growth is Girsanov theorem. Suppose d = 1 for simplicity.
Let us consider an (Ft )-classical Brownian motion (Xt ). We set
$ t
b(s, Xs )ds.
Wt = Xt −
0
Under some suitable probability Q, (Wt ) is an (Ft )- classical Brownian
motion. Therefore (Xt ) provides a solution to E(1, b; δ0 ).
We continue with an example where E(a, b) does not admit pathwise
uniqueness, even though it admits uniqueness in law.
Example 4.7 We consider the stochastic equation
$ t
Xt =
sign(Xs )dWs .
0
with
sign(x) =
"
1
if x ≥ 0,
−1 if x < 0,
It corresponds to E(a, b; δ0 ) with b = 0 and a(x) = sign(x).
(4.12)
Stochastic Differential Equations.
12
If (Wt )t≥0 is an (Ft )- classical Brownian motion then (Xt )t≥0 is (Ft )t≥0 continuous local martingale vanishing at zero such that [X]t ≡ t. Ac-
cording to Lévy caracterization theorem stated in section 2, X is an
(Ft )t≥0 -classical Brownian motion. This shows in particular that E(a, b; δ0 )
admits uniqueness in law. In the sequel we will show that E(a, b; δ0 )
also admits weak existence.
Let now (Ω, F , P) be a probability space, an (Ft )t≥0 - classical Brown-
ian motion with respect to a filtration and (Xt )t≥0 such that (4.12) is
verified. Then X̃t = −Xt can be shown to be also a solution. Therefore
E(a, b; δ0 ) does not admit pathwise uniqueness.
We continue stating a result true in the multi-dimensional case.
Proposition 4.8 (Stroock-Varadhan). Let ν be a probability on Rd
such that
$
R
&x&2m ν(dx) < +∞,
(4.13)
for a certain m > 1. We suppose that a, b are continuous with linear
growth. Then E(a, b; ν) admits weak existence.
From now on, a function γ : [0, T ] × Rm → Rd will be said Hölder
continuous if it is Hölder continuous in the space variable x ∈ Rm
uniformly with respect to the time variable t ∈ [0, T ].
From the same authors, an easy consequence of Th. 7.2.1 in [18] is the
following result.
Proposition 4.9 We suppose a, b both Hölder continuous, bounded
such that conditon (3.8) is fulfilled. Then SDE E(a, b; ν) admits weak
uniqueness.
Remark 4.10
(i) The Hölder condition and (3.8) in Proposition 4.9
may be relaxed and replaced with the solvability of a Cauchy problem of a parabolic PDE with suitable terminal value.
Stochastic Differential Equations.
13
(ii) In the case d = 1, if a, b are bounded and just Borel with (3.8)
for x on each compact, then E(a, b; ν) admits weak existence and
uniqueness in law. See [18] exercises 7.3.2 and 7.3.3.
(iii) If d = 2, the same holds as at previous point provided that moreover a does not depend on time.
We proceed with some more specifically unidimensional material stating
some results from K.J. Engelbert and W. Schmidt. These two authors
furnished necessary and sufficient conditions for weak existence and
uniqueness in law of SDEs.
For a Borel function σ : R → R, we first define
Z(σ) = {x ∈ R|σ(x) = 0};
then we define the set I(σ) as the set of real numbers x such that
$ x+ε
dy
= ∞, ∀ε > 0.
2
x−ε σ (y)
Proposition 4.11 (Engelbert-Schmidt criterion).
Suppose that a : R → R, i.e. does not depend on time and we consider
the equation without drift E(a, 0).
i) E(a, 0) admits weak existence (without explosion) if and only if
I(a) ⊂ Z(a)
(4.14)
ii) E(a, 0) admits weak existence and uniqueness in law if and only if
I(a) = Z(a)
(4.15)
Remark 4.12 i) If a is continuous then (4.14) is always verified.
Indeed, if a(x) /= 0, there is ε > 0 such that
|a(y)| > 0, ∀y ∈ [x − ε, x + ε].
Therefore x cannot belong to I(a).
Stochastic Differential Equations.
14
ii) (4.14) is verified also for some discontinuous functions as for instance a(x) = sign(x). This confirms what was affirmed previously, i.e. the weak existence (and uniqueness in law) for E(a, 0).
iii) If a(x) = 1{0} (x), (4.14) is not verified.
iv) If a(x) = |x|α , α ≥
1
2
then
Z(a) = I(a) = {0}.
So there is at most one solution in law for E(a, 0).
v) The proof is technical and makes use of Lévy characterisation theorem of Brownian motion.
4.2
Results on pathwise uniqueness
Proposition 4.13 (Yamada-Watanabe)
Let a, b : R+ × R → R and consider again E(a, b). Suppose b globally
Lipschitz and h : R+ → R+ strictly increasing continuous such that
• h(0) = 0;
$ ε
1
(y)dy = ∞, ∀ε > 0;
•
2
0 h
• |a(t, x) − a(t, y)| ≤ h(x − y).
Then pathwise uniqueness is verified.
Remark 4.14
uα ,
α > 12 .
• In Proposition 4.13, one typical choice is h(u) =
• Pathwise uniqueness for E(a, b) holds therefore if b is globally
Lipschitz and a is Hölder continuous with parameter equal to 12 .
Stochastic Differential Equations.
15
Corollary 4.15 Suppose that the assumption of Proposition 4.13 are
verified and a, b continuous with linear growth. Then E(a, b; ν) admits
strong existence and pathwise uniqueness, whenever ν verifies condition
(4.13).
Proof. It follows from Propositions 4.13 and 4.8 together with Proposition 4.5 ii).
Remark 4.16 Suppose d = 1. Pathwise uniqueness for E(a, b) also
holds under the following assumptions.
(i) a, b are bounded, a is time-indepedent and a ≥ const > 0, h as in
Proposition 4.13. This result has an analogous form in the case of
space-time white noise driven SPDEs of parabolic type, as proved
by Bally, Gyongy and Pardoux in 1994.
(ii) a independent on time, b bounded and a ≥ const > 0; moreover
|a(x) − a(y)|2 ≤ |f (y) − f (x)| and f is increasing and bounded.
For illustration we provide some significant examples.
Example 4.17
Xt =
$
t
0
|Xs |α dWs , t ≥ 0.
(4.16)
We set a(x) = |x|α , 0 < α < 1. This is equation E(a, 0) with a(x) =
|x|α . According to Engelbert-Schmidt notations, we have Z(a) = {0}.
Moreover
• If α ≥
1
2
then I(a) = {0}.
• If α <
1
2
then I(a) = ∅.
Therefore according to Proposition 4.11, E(a, 0) admits weak existence.
On the other hand, if α ≥ 12 ,
|xα − y α | ≤ h(|x − y|),
(4.17)
Stochastic Differential Equations.
16
where h(z) = z α . According to Proposition 4.13, (4.16) admits pathwise
uniqueness and by Corollary 4.15, also strong existence. The unique
solution is X ≡ 0.
If α < 12 , X ≡ 0 is always a solution. This is not the only one; even
uniqueness in law is not true.
Example 4.18 Let a(x) =
!
|x|, b Lipschitz.
Then E(a, b) admits strong existence and pathwise uniqueness.
In fact, a is Hölder continuous with parameter
1
2
and the second item
of Remark 4.14 applies; so pathwise uniqueness holds. Strong existence
is a consequence of Propositions 4.8 and 4.5 ii).
An interesting particular case is provided by the following equation.
Let x0 , σ, δ ≥ 0, k ∈ R. The following equation admits strong existence
and pathwise uniqueness.
$ t!
$ t
|Zs |dWs +
(δ − kXs )ds,
Zt = x0 + σ
0
0
t ∈ [0, T ].
(4.18)
Equation (4.18) is widely used in mathematical finance and it constitutes the model of Cox-Ingersoll-Ross: the solution of the mentioned
equation represents the short interest rate.
Consider now the particular case where k = 0, σ = 2. According to some
comparison theorem for SDEs, the solution Z is always non-negative
and therefore the absolute value may be omitted. The equation becomes
Zt = x0 + 2
$ t!
Zs dWs + δt.
(4.19)
Definition 4.19 The unique solution Z to
$ t!
Zs dWs + δt
Zt = x0 + 2
(4.20)
0
0
is called square δ-dimensional Bessel process starting at x0 ; it is
denoted by BESQδ (x0 ); for fine properties of this process, see [17], ch.
IX.3.
Stochastic Differential Equations.
17
Since Z ≥ 0, we call δ-dimensional Bessel process starting from x0
√
the process X = Z. It is denoted by BES δ (x0 ).
Remark 4.20 Let d ≥ 1. Let W = (W 1 , · · · , W d ) be a classical
d−dimensional Brownian motion. We set Xt = &Wt &. (Xt )t≥0 is a
d-dimensional Bessel process.
Remark 4.21 If δ > 1, it is possible to see that
$
δ − 1 t ds
Xt = Wt +
.
2
0 Xs
4.3
The case with distributional drift
Pioneering work about diffusions with generalized drift was presented
by N.I. Portenko, but in the framework of semimartingale processes.
Recently some work was done caracterizing solutions in the class of the
so called Dirichlet processes, with some motivations in random irregular
environment.
An useful transformation in the theory of stochastic differential equation is the so called Zvonkin transformation. Let (Wt ) be an (Ft )classical Brownian motion. Let a, b : R → R respectively C 1 and locally
bounded. We suppose moreover a > 0. We fix x0 ∈ R. Let (Xt )t≥0 be
a solution of
Xt = x0 +
We set Σ(x) =
$
$
t
b(Xs )ds +
0
x
0
$
t
a(Xs )dWs .
(4.21)
0
2b
(y)dy and we define h : R → R such that
a2
h(0) = 0,
h' = e−Σ .
h is strictly increasing. We set ã(x) = (ah' )(h−1 (x)) where h−1 is the
inverse of h. We set Yt = h(Xt ). Without entering into details, the
classical Itô formula allows to show that (Yt ) defines a solution of
"
dYt = ã(Yt )dWt
(4.22)
Y0 = h(x0 ).
Stochastic Differential Equations.
18
Now, equation (4.22) fulfills the requirements of Engelbert-Schmidt criterion so that it admits weak existence and uniqueness in law. Consequently, unless explosion, one can easily establish the same wellposedness for (4.21).
Zvonkin transformation also allows to prove strong existence and pathwise uniqueness results for (4.21; for instance when
• a has linear growth;
• y→
#y
b(s)
ds
0 a2 (s)
is a bounded function.
In fact, problem (4.22) satisfies pathwise uniqueness and strong existence since the coefficients are Lipschitz with linear growth. Therefore
one can deduce the same for (4.21).
Veretennikov generalized Zvonkin transformation to the d-dimensional
case in some cases which include the cas a = 1 and b bounded Borel.
Zvonkin’s procedure suggests also to consider a formal equation of the
type
dXt = dWt + γ ' (Xt )dt,
(4.23)
where γ is only a continuous function and so b = γ ' is a Schwartz
distribution; γ could be for instance the realization of an independent
Brownian motion of W . Therefore equation (4.23) is motivated by
the study of irregular random media. When σ = 1, b = γ ' , stochastic
differential equation (4.22), h' = e−2γ still makes sense.
Using Engelbert-Schmidt criterion, one can see that problem (4.22) still
admits weak existence and uniqueness in the sense of distribution laws.
If Y is a solution of (4.22), X = h−1 (Y ) provides a natural candidate
solution for (4.21). R.F. Bass, Z-Q. Chen and F. Flandoli, F. Russo, J.
Wolf investigated generalized SDEs as (4.23): in particular they made
previous reasonning rigorous, respectively in the case of strong and
weak solutions, see [5].
Stochastic Differential Equations.
5
19
Connected topics
We aim here at giving some basic references about topics which are
closely connected to stochastic differential equations.
• Stochastic partial differential equations (SPDEs)
If a stochastic differential equation is a random perturbation of
an ordinary differential equation, a stochastic partial differential
equation is a random perturbation of a partial differential equation. Big activity was performed in the parabolic (evolution equation) and hyperbolic case (wave equation). Most of the work was
done in the case of a fixed underlying probability spaces. We only
quote two basic monographies which should be consulted at first
before getting into the subject: the one of J. Walsh [19] and the
one of G. Da Prato and J. Zabczyk [4].
However it was possible to establish some results about weak existence and uniqueness in law for SPDEs. On possible tool was a
generalization of Girsanov theorem to the case of Gaussian spacetime white noise. Weak existence for the stochastic quantization
equation was proved with the help of infinite dimensional Dirichlet forms by S. Albeverio and M. Röckner.
We also indicate a beautiful recent monography by G. Da Prato
which pays particular attention to Kolmogorov equations with
infinitely many variables, see [3].
• Numerical approximations
Relevant work was done in numerical approximation of solutions
to SDEs and related approximations of solutions to linear parabolic
equations via Feynmann-Kac probabilistic representation, see Theorem 3.3). It seems that the stochastic simulations (of improved
Monte Carlo type and related topics) for solving deterministic
problems are efficient when the space dimension is greater than
4.
Stochastic Differential Equations.
20
• Malliavin calculus
Malliavin calculus is a wide topic which is the object of another
article in this Encyclopedia. Relevant applications of it concern
stochastic (ordinary and partial) differential equations. We only
quote a monography of D. Nualart on those applications, [12].
Two main objects were studied.
– Given a solution of a stochastic differential equation (Xt ),
sufficient conditions so that the law of Xt , t > 0, has a
(smooth) density p(t, ·). Small time asymptotics of this den-
sity, when t → 0, and small drift perturbation were performed, refining Freidlin-Ventsell large deviation estimates.
– Coming back to SDE (4.11), one can conceive to consider
coefficients a, b non adapted with respect to the underlying
filtration (Ft ). On the other hand, the initial condition ξ
may be anticipating, i.e. not F0 -measureble. In that case
#t
the Itô integral 0 a(s, Xs )dWs is not defined. A replacement
tool is the so called Skorohod integral.
• Rough paths approach
A very successful and significant research field is the rough path
theory. In the case of dimension d = 1, Doss-Sussmann method
allows to transform the solution of a stochastic differential equation into the solution of an ordinary (random) differential equation. In particular, that solution can be seen as depending (pathwise) continuously from the driving Brownian motion (Wt ) with
respect to the usual topology of C([0, T ]). Unless exceptions,
this continuity does not hold in case of general dimension d > 1.
Rough paths theory, introduced by T. Lyons, allows to recover
somehow this lack of continuity and establishes a true pathwise
stochastic integration.
• SDEs driven by non-semimartingales
Stochastic Differential Equations.
21
At the present moment, there is a very intense activity towards
SDEs driven by processes which are not semimartingales. In
this perspective, we list SDEs driven by fractional Brownian motion with the help of rough paths theory, using fractional and
Young type integrals and involving finite cubic variation processes. Among the contributors in that area we quote L. Coutin,
R. Coviello, M. Errami, M. Gubinelli, Z. Qian, F. Russo, P. Vallois, M. Zähle.
KEY WORDS
Stochastic ordinary (partial) differential equations, Feynmann-Kac formula, probabilistic representation of a (linear or non-linear) PDE, Brownian motion, semimartingales, diffusion process, pathwise uniqueness,
strong existence, weak existence, uniqueness in law, Peano phenomena, Girsanov theorem, Bessel process, Malliavin calculus, Skorohod
integral.
SEE ALSO
63 RANDOM WALK IN RANDOM ENVIRONMENTS
137 INTERACTING STOCHASTIC PARTICLE SYSTEMS
372 RANDOM DYNAMICAL SYSTEMS
463 MALLIAVIN CALCULUS
477 STOCHASTIC HYDRODYNAMICS
Stochastic Differential Equations.
22
References
[1] Benachour, S., Chassaing, P., Roynette, B., Vallois, P. Processus
associés l’ équation des milieux poreux. Ann. Scuola Norm. Sup.
Pisa Cl. Sci. (4) 23 (1996), no. 4, 793–832 (1997).
[2] Bouleau, N., Lépingle, D., Numerical methods for stochastic processes. John Wiley, 1994.
[3] Da Prato, G., Kolmogorov equations for stochastic PDEs.
Birkäuser, 2004.
[4] Da Prato, G., Zabczyk, J., Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992.
[5] Flandoli, F., Russo, F., Wolf, J. Some stochastic differential equations with distributional drift. Part I: General Calculus. Osaka
Journal of Mathematics. Vol. 40, No 2, 493-542 (2003).
[6] Fleming, W.H., Soner, M. Controlled Markov processes and viscosity solutions. Springer-Verlag 1993.
[7] Graham, C., Kurtz, Th. G., Méléard, S., Protter, Ph. E., Pulvirenti, M., Talay, D. Probabilistic models for nonlinear partial differential equations. Lectures given at the 1st Session and Summer
School held in Montecatini Terme, May 22–30, 1995. Edited by Talay and L. Tubaro. Lecture Notes in Mathematics, 1627. SpringerVerlag. Centro Internazionale Matematico Estivo (C.I.M.E.), Florence, 1996.
[8] Kloeden, P.E., Platen, E. Numerical solutions of stochastic differential equations. Springer-Verlag, 1992.
[9] Karatzas, I., Shreve, S.E., Brownian Motion and Stochastic Calculus. Springer Verlag, Second edition, 1991.
Stochastic Differential Equations.
23
[10] Lamberton, D., Lapeyre, B., Introduction au calcul stochastique
et applications la finance. Collection Ellipses, 1997.
[11] Ma, J., Yong, J. Forward-backward stochastic differential equations and their applications. Lecture Notes in Mathematics, 1702.
Springer-Verlag, 1999.
[12] Nualart, D., The Malliavin Calculus and Related Topics. Springer
Verlag, 1995.
[13] Øksendal, B., Stochastic differential equations, Springer-Verlag.
Fifth Edition, 1998.
[14] Pardoux, E., Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs
of second order. Stochastic analysis and related topics, VI (Geilo,
1996), 79–127, Progr. Probab., 42, Birkäuser, 1998.
[15] Protter, P., Stochastic Integration and Differential Equations.
Springer-Verlag, 1992.
[16] Lyons, T., Qian, Z. Controlled systems and rough paths. Oxford
University Press, Oxford, 2002.
[17] Revuz, D., Yor, M., Continuous Martingales and Brownian Motion. Springer-Verlag, 1994.
[18] Stroock, D., Varadhan, Multidimensional diffusion processes.
Springer-Verlag, 1979.
[19] Walsh, J.B., An introduction to stochastic partial differential equations. Ecole d’été de probabilités de Saint-Flour, XIV—1984, 265–
439, Lecture Notes in Math., 1180, Springer-Verlag, 1986.
[20] Yong, J., Zhou, X. Y., Stochastic controls: Hamiltonian systems
and HJB equations. Springer-Verlag, 1999.