Download Time-change equations for diffusion processes • Weak and strong

Time-change equations for diffusion processes
• Weak and strong solutions for simple stochastic equations
• Equivalence of notions of uniqueness
• Compatibility restrictions
• Convex constraints
• Ordinary stochastic differential equations
• The Yamada-Watanabe and Engelbert theorems
• Stochastic equations for Markov chains
• Diffusion limits??
• Uniqueness question
• Compatibility for multiple time-changes
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Weak and strong solutions for simple stochastic equations
Given measurable Γ : S1 × S2 → R and a S2 -valued random variable
Y , consider the equation
Γ(X, Y ) = 0.
(1)
In many (most?) contexts, it is natural to specify the distribution
ν ∈ P(S2 ) of Y rather than a particular Y on a particular probability
space.
Following the terminology of Engelbert (1991) and Jacod (1980), we
refer to the joint distribution of (X, Y ) as a joint solution measure. In
particular, µ is a joint solution measure if µ(S1 × ·) = ν and
Z
|Γ(x, y)|µ(dx × dy) = 0.
S1 ×S2
(Without loss of generality, we can assume that Γ is bounded.)
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Strong solutions
Definition 1 A solution (X, Y ) for (Γ, ν) is a strong solution if there
exists a Borel measurable function F : S2 → S1 such that X = F (Y ) a.s.
If a strong solution exists on some probability space, then a strong
solution exists for any Y with distribution ν.
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Disintegration of measures
Lemma 2 If µ ∈ P(S1 × S2 ) and µ(S1 × ·) = ν, then there exists a transition function η such that µ(dx × dy) = η(y, dx)ν(dy).
There exists G : S2 × [0, 1] → S1 such that if Y has distribution ν and ξ
is independent of Y and uniformly distributed on [0, 1], (G(Y, ξ), Y ) has
distribution µ.
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Joint solution measures for strong solutions
Let SΓ,ν ⊂ P(S1 ×S2 ) denote the collection of joint solution measures.
Clearly, SΓ,ν is convex, and if Γ is continuous, then SΓ,ν is closed in
the weak topology.
Lemma 3 If µ ∈ SΓ,ν , then µ corresponds to a strong solution if and only if
there exists a Borel measurable F : S2 → S1 , such that η(y, dx) = δF (y) (dx)
a.s. ν.
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Notions of uniqueness
Definition 4 Pointwise uniqueness holds, if X1 , X2 , and Y defined on
the same probability space with µX1 ,Y , µX2 ,Y ∈ SΓ,ν implies X1 = X2 a.s.
For µ ∈ SΓ,ν , µ-pointwise uniqueness holds if X1 , X2 , and Y defined on
the same probability space with µX1 ,Y = µX2 ,Y = µ implies X1 = X2 a.s.
Joint uniqueness in law (or weak joint uniqueness) holds, if SΓ,ν contains at most one measure.
Uniqueness in law (or weak uniqueness) holds if all µ ∈ SΓ,ν have the
same marginal distribution on S1 .
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Strong solutions and pointwise uniqueness
Remark 5 If µ ∈ SΓ,ν corresponds to a strong solution, then µ-pointwise
uniqueness holds.
Lemma 6 If every solution is a strong solution, then pointwise uniqueness
holds.
Proof. Let G1 and G2 be functions corresponding to strong solutions
and define
G1 (y)
u > 1/2
G3 (y, u) =
G2 (y)
u ≤ 1/2.
Then for Y and ξ independent, µY = ν and ξ uniform on [0, 1],
Γ(G3 (Y, ξ), Y ) = Γ(G1 (Y ), Y )1{ξ>1/2} + Γ(G2 (Y ), Y )1{ξ≤1/2} = 0,
and hence G3 (Y, ξ) is a solution.
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Equivalence of notions of uniqueness
Proposition 7 The following are equivalent:
a) Pointwise uniqueness.
b) µ-pointwise unqueness for every µ ∈ SΓ,ν .
c) Joint uniqueness in law.
d) Uniqueness in law.
Kurtz (2007)
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Temporal compatibility restrictions
Let E1 and E2 be Polish spaces and let DEi [0, ∞), be the Skorohod
space of cadlag Ei -valued functions. Let Y be a process in DE2 [0, ∞).
By FtY , we mean σ(Y (s), s ≤ t).
Definition 8 A process X in DE1 [0, ∞) is compatible with Y if for each
t ≥ 0 and h ∈ B(DE2 [0, ∞)),
E[h(Y )|FtX,Y ] = E[h(Y )|FtY ] a.s.
(2)
If Y has independent increments, then X is compatible with Y if
Y (t + ·) − Y (t) is independent of FtX,Y for all t ≥ 0.
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General compatibility restrictions
If BαS1 is a sub-σ-algebra of B(S1 ) and X is an S1 -valued random variable on (Ω, F, P ), then FαX ≡ {{X ∈ D} : D ∈ BαS1 } is the sub-σalgebra of F generated by {h(X) : h ∈ B(BαS1 )}, where B(BαS1 ) is the
collection of h ∈ B(S1 ) that are BαS1 -measurable.
Definition 9 Let A be an index set and for each α ∈ A, let BαS1 be a sub-σalgebra of B(S1 ) and BαS2 be a sub-σ-algebra of B(S2 ). Let Y be an S2 -valued
random variable. An S1 -valued random variable X is compatible with Y
if for each α ∈ A and each h ∈ B(S2 ),
E[h(Y )|FαX ∨ FαY ] = E[h(Y )|FαY ] a.s.,
(3)
where FαX ≡ {{X ∈ D} : D ∈ BαS1 } and FαY ≡ {{Y ∈ D} : D ∈
BαS2 }. The collection C ≡ {(BαS1 , BαS2 ) : α ∈ A} will be refered to as a
compatibility structure.
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Compatibility is a distributional property
Note that (3) is equivalent to requiring that for each h ∈ B(S2 ),
inf
S
S
f ∈B(Bα1 ×Bα2 )
E[(h(Y ) − f (X, Y ))2 ] =
inf S E[(h(Y ) − f (Y ))2 ],
(4)
f ∈B(Bα2 )
so compatibility is a property of the joint distribution of (X, Y ).
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Convexity
Lemma 10 Let C be a compatibility structure and ν ∈ P(S2 ). Let SC,ν be
the collection of µ ∈ P(S1 × S2 ) with the following properties:
a) µ(S1 × ·) = ν
b) If (X, Y ) has distribution µ, then X is C-compatible with Y .
Then SC,ν is convex.
Proof. Note that the right side of (4) is determined by ν, so µ ∈ SC,ν
if µ(S1 × ·) = ν and
Z
Z
(h(y) − f (x, y)2 µ(dx × dy) ≥ inf S
(h(y) − g(y))2 ν(dy),
S1 ×S2
g∈B(Bα2 )
S2
for each h ∈ B(S2 ), each α ∈ A, and each f ∈ B(BαS1 × BαS2 ).
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Convex constraints
Γ denotes a collection of constraints that determine convex subsets
of P(S1 × S2 ), and SΓ,C,ν = {µ ∈ SC,ν : µ satisfies Γ}
For example, finiteness and moment conditions
h(X, Y ) < ∞
a.s. or E[|h(X, Y )|] < ∞
inequalities
h(X, Y ) ≤ g(X, Y ) a.s. or E[h(X, Y )] ≤ E[g(X, Y )]
equations
h(X, Y ) = 0 a.s.
limit requirements
lim E[|hn (X, Y )|] = 0
n→∞
optimality conditions
Z
E[h(X, Y )] =
inf
µ∈SΓ0 ,C,ν
hdµ
S1 ×S2
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Ordinary stochastic differential equations
U a process in DRd [0, ∞)
V an Rm -valued semimartingale with respect to the filtration {FtU,V }
H : DRd [0, ∞) → DMd×m [0, ∞) Borel measurable satisfying H(x, t) =
H(x(· ∧ t), t).
Then X is a solution of
Z
X(t) = U (t) +
t
H(X, s−)dV (s)
0
if X is compatible with Y = (U, V ) and
X
k+1
k
k
∧t)−V ( ∧t))|] = 0,
lim E[1∧|X(t)−U (t)−
H(X, )(V (
n→∞
n
n
n
t ≥ 0.
k
But see Karandikar (1995).
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Pointwise uniqueness with compatibility constraints
X1 , X2 , and Y defined on the same probability space.
X1 and X2 , S1 -valued and Y S2 -valued.
(X1 , X2 ) are jointly compatible with Y if
E[f (Y )|FαX1 ∨ FαX2 ∨ FαY ] = E[f (Y )|FαY ],
α ∈ A, f ∈ B(S2 ).
Pointwise uniqueness holds for compatible solutions of (Γ, ν), if for every triple of processes (X1 , X2 , Y ) defined on the same sample space
such that µX1 ,Y , µX2 ,Y ∈ SΓ,C,ν and (X1 , X2 ) is jointly compatible with
Y , X1 = X2 a.s.
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The Yamada-Watanabe and Engelbert theorems
SΓ,C,ν denotes the convex subset of µ ∈ P(S1 × S2 ) such that µ fulfills
the constraints in Γ, µ is C-compatible, and µ(S1 × ·) = ν.
Theorem 11 Suppose SΓ,C,ν 6= ∅. The following are equivalent:
a) Pointwise uniqueness holds for compatible solutions.
b) Joint uniqueness in law holds for compatible solutions and there exists
a strong, compatible solution.
Theorem 12 Let µ ∈ SΓ,C,ν . Then µ-pointwise uniqueness holds if and
only if the solution corresponding to µ is strong.
If µ-pointwise uniqueness holds for every µ ∈ SΓ,C,ν , then every solution is
strong and pointwise uniqueness holds.
Kurtz (2007) cf. Yamada and Watanabe (1971), Engelbert (1991)
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Stochastic equations for Markov chains
Specify a continuous time Markov chain by specifying the intensities
of its possible jumps
P {X(t + ∆t) = X(t) + ζk |FtX } ≈ βk (X(t))∆t
Given the intensities, the Markov chain satisfies
Z t
X
X(t) = X(0) +
Yk ( βk (X(s))ds)ζk
k
0
where the Yk are independent unit Poisson processes.
(Assume that there are only finitely many ζk .)
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Diffusion limits??
Possible scaling limits of the form
Z
Z t
1Xe 2 t n
Yk (n
βk (Xn (s))ds)ζk +
F n (Xn (s))ds
Xn (t) = Xn (0) +
n
0
0
k
where Yek (u) = Yk (u) − u and F n (x) =
P
k
nβkn (x)ζk .
Note that n1 Yek (n2 ·) ≈ Wk
Assuming Xn (0) → X(0), βkn → βk and F n → F , we might expect a
limit satisfying
Z t
Z t
X
F (X(s))ds
X(t) = X(0) +
Wk ( βk (X(s))ds)ζk +
k
0
0
Kurtz (1980), Ethier and Kurtz (1986), Section6.5.
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Uniqueness question
X(t) = X(0) +
X
k
Let τk (t) =
Z t
Z t
Wk ( βk (X(s))ds)ζk +
F (X(s))ds
0
0
Rt
Rt
β
(X(s))ds
and
γ(t)
=
k
0
0 F (X(s))ds. Then
X
τ̇l (t) = βl (X(0) +
Wk (τk (t))ζk + γ(t))
γ̇(t) = F (X(0) +
k
X
Wk (τk (t))ζk + γ(t))
k
Problem: Find conditions under which pathwise uniqueness holds.
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Compatibility for multiple time-changes
X(t) = X(0) +
m
X
k=1
Set τk (t) =
Rt
0
FαY
Z t
Z t
Wk ( βk (X(s))ds)ζk +
F (X(s))ds
0
0
βk (X(s))ds, and for α ∈ [0, ∞)m , define
= σ(Wk (sk ) : sk ≤ αk , k = 1, 2, . . .) ∨ σ(X(0))
and
FαX = σ({τ1 (t) ≤ s1 , τ2 (t) ≤ s2 , . . .} : si ≤ αi , i = 1, 2, . . . , t ≥ 0).
Rt
If X is a compatible solution, then Wk ( 0 βk (X(s))ds), k = 1, . . . , m
are martingales with respect to the same filtration and hence X is a
solution of the martingale problems for
1X
aij (x)∂i ∂j f (x) + F (x) · ∇f (x),
Af (x) =
2 i,j
P
a(x) = k βk (x)ζk ζkT .
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Two-dimensional case
For i = 1, 2, Wi
standard Brownian motions.
βi : R2 → (0, ∞), bounded
Z t
X1 (t) = W1 ( β1 (X(s))ds)
Z t
X2 (t) = W2 ( β2 (X(s))ds)
0
0
or equivalently
τ̇i (t) = βi (W1 (τ1 (t)), W2 (τ2 (t))),
i = 1, 2
A strong, compatible solution exists, and weak uniqueness holds by
Stroock-Varadhan, so pathwise uniqueness holds.
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References
H. J. Engelbert. On the theorem of T. Yamada and S. Watanabe. Stochastics Stochastics Rep., 36(3-4):205–216,
1991. ISSN 1045-1129.
Stewart N. Ethier and Thomas G. Kurtz. Markov processes: Characterization and Convergence. Wiley Series in
Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc.,
New York, 1986. ISBN 0-471-08186-8.
Jean Jacod. Weak and strong solutions of stochastic differential equations. Stochastics, 3(3):171–191, 1980. ISSN
0090-9491.
Rajeeva L. Karandikar. On pathwise stochastic integration. Stochastic Process. Appl., 57(1):11–18, 1995. ISSN
0304-4149.
Thomas G. Kurtz. Representations of Markov processes as multiparameter time changes. Ann. Probab., 8(4):
682–715, 1980. ISSN 0091-1798.
Thomas G. Kurtz. The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities.
Electron. J. Probab., 12:951–965, 2007. ISSN 1083-6489. doi: 10.1214/EJP.v12-431. URL http://dx.doi.
org/10.1214/EJP.v12-431.
Toshio Yamada and Shinzo Watanabe. On the uniqueness of solutions of stochastic differential equations. J.
Math. Kyoto Univ., 11:155–167, 1971.
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Abstract
Time-change equations for diffusion processes
General notions of weak and strong solutions of stochastic equations will be described and a general version
of the Yamada-Watanabe-Engelbert theorem relating existence and uniqueness of weak and strong solutions
given. Time-change equations for diffusion processes provide an interesting example. Such equations arise
naturally as limits of analogous equations for Markov chains. For one-dimensional diffusions they also are
essentially given in the now-famous notebook of Doeblin. Although requiring nothing more than standard
Brownian motions and the Riemann integral to define, the question of strong uniqueness remains unresolved.
To prove weak uniqueness, the notion of compatible solution is introduced and the martingale properties of
compatible solutions used to reduce the uniqueness question to the corresponding question for a martingale
problem or an Ito equation.
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