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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


Weak and strong solutions for simple stochastic equa-tions

Given measurable Γ : S1 × S2 → R and a S2-valued random variableY , consider the equation

Γ(X, Y ) = 0. (1)

In many (most?) contexts, it is natural to specify the distributionν ∈ P(S2) of Y rather than a particular Y on a particular probabilityspace.

Following the terminology of Engelbert (1991) and Jacod (1980), werefer to the joint distribution of (X, Y ) as a joint solution measure. Inparticular, µ is a joint solution measure if µ(S1 × ·) = ν and∫

S1×S2

|Γ(x, y)|µ(dx× dy) = 0.

(Without loss of generality, we can assume that Γ is bounded.)


Strong solutions

Definition 1 A solution (X, Y ) for (Γ, ν) is a strong solution if thereexists 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 strongsolution exists for any Y with distribution ν.


Disintegration of measures

Lemma 2 If µ ∈ P(S1× S2) and µ(S1× ·) = ν, then there exists a transi-tion 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 ξis independent of Y and uniformly distributed on [0, 1], (G(Y, ξ), Y ) hasdistribution µ.


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 inthe weak topology.

Lemma 3 If µ ∈ SΓ,ν , then µ corresponds to a strong solution if and only ifthere exists a Borel measurable F : S2 → S1, such that η(y, dx) = δF (y)(dx)a.s. ν.


Notions of uniqueness

Definition 4 Pointwise uniqueness holds, if X1, X2, and Y defined onthe 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 onthe 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 con-tains at most one measure.

Uniqueness in law (or weak uniqueness) holds if all µ ∈ SΓ,ν have thesame marginal distribution on S1.


Strong solutions and pointwise uniqueness

Remark 5 If µ ∈ SΓ,ν corresponds to a strong solution, then µ-pointwiseuniqueness holds.

Lemma 6 If every solution is a strong solution, then pointwise uniquenessholds.

Proof. Let G1 and G2 be functions corresponding to strong solutionsand define

G3(y, u) =

G1(y) u > 1/2G2(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 )1ξ>1/2 + Γ(G2(Y ), Y )1ξ≤1/2 = 0,

and hence G3(Y, ξ) is a solution.


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)


Temporal compatibility restrictions

Let E1 and E2 be Polish spaces and let DEi[0,∞), be the Skorohodspace of cadlag Ei-valued functions. Let Y be a process in DE2

[0,∞).By FY

t , 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 )|FX,Yt ] = E[h(Y )|FY

t ] a.s. (2)

If Y has independent increments, then X is compatible with Y ifY (t+ ·)− Y (t) is independent of FX,Y

t for all t ≥ 0.


General compatibility restrictions

If BS1α is a sub-σ-algebra of B(S1) and X is an S1-valued random vari-

able on (Ω,F , P ), then FXα ≡ X ∈ D : D ∈ BS1

α is the sub-σ-algebra of F generated by h(X) : h ∈ B(BS1

α ), where B(BS1α ) is the

collection of h ∈ B(S1) that are BS1α -measurable.

Definition 9 Let A be an index set and for each α ∈ A, let BS1α be a sub-σ-

algebra of B(S1) and BS2α 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 Yif for each α ∈ A and each h ∈ B(S2),

E[h(Y )|FXα ∨ FY

α ] = E[h(Y )|FYα ] a.s., (3)

where FXα ≡ X ∈ D : D ∈ BS1

α and FYα ≡ Y ∈ D : D ∈

BS2α . The collection C ≡ (BS1

α ,BS2α ) : α ∈ A will be refered to as a

compatibility structure.


Compatibility is a distributional property

Note that (3) is equivalent to requiring that for each h ∈ B(S2),

inff∈B(BS1

α ×BS2α )

E[(h(Y )− f(X, Y ))2] = inff∈B(BS2

α )

E[(h(Y )− f(Y ))2], (4)

so compatibility is a property of the joint distribution of (X, Y ).


Convexity

Lemma 10 Let C be a compatibility structure and ν ∈ P(S2). Let SC,ν bethe 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,νif µ(S1 × ·) = ν and∫

S1×S2

(h(y)− f(x, y)2µ(dx× dy) ≥ infg∈B(BS2

α )

∫S2

(h(y)− g(y))2ν(dy),

for each h ∈ B(S2), each α ∈ A, and each f ∈ B(BS1α × BS2

α ).


Convex constraints

Γ denotes a collection of constraints that determine convex subsetsof 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 )]

equationsh(X, Y ) = 0 a.s.

limit requirementslimn→∞

E[|hn(X, Y )|] = 0

optimality conditions

E[h(X, Y )] = infµ∈SΓ0,C,ν

∫S1×S2

hdµ


Ordinary stochastic differential equations

U a process in DRd[0,∞)

V an Rm-valued semimartingale with respect to the filtration FU,Vt

H : DRd[0,∞) → DMd×m[0,∞) Borel measurable satisfying H(x, t) =H(x(· ∧ t), t).

Then X is a solution of

X(t) = U(t) +

∫ t

0

H(X, s−)dV (s)

if X is compatible with Y = (U, V ) and

limn→∞

E[1∧|X(t)−U(t)−∑k

H(X,k

n)(V (

k + 1

n∧t)−V (

k

n∧t))|] = 0, t ≥ 0.

But see Karandikar (1995).


Pointwise uniqueness with compatibility constraintsX1, 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 )|FX1α ∨ FX2

α ∨ FYα ] = E[f(Y )|FY

α ], α ∈ A, f ∈ B(S2).

Pointwise uniqueness holds for compatible solutions of (Γ, ν), if for ev-ery triple of processes (X1, X2, Y ) defined on the same sample spacesuch that µX1,Y , µX2,Y ∈ SΓ,C,ν and (X1, X2) is jointly compatible withY , X1 = X2 a.s.


The Yamada-Watanabe and Engelbert theoremsSΓ,C,ν denotes the convex subset of µ ∈ P(S1 × S2) such that µ fulfillsthe 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 existsa strong, compatible solution.

Theorem 12 Let µ ∈ SΓ,C,ν . Then µ-pointwise uniqueness holds if andonly if the solution corresponding to µ is strong.

If µ-pointwise uniqueness holds for every µ ∈ SΓ,C,ν , then every solution isstrong and pointwise uniqueness holds.

Kurtz (2007) cf. Yamada and Watanabe (1971), Engelbert (1991)


Stochastic equations for Markov chainsSpecify a continuous time Markov chain by specifying the intensitiesof its possible jumps

PX(t+ ∆t) = X(t) + ζk|FXt ≈ βk(X(t))∆t

Given the intensities, the Markov chain satisfies

X(t) = X(0) +∑k

Yk(

∫ t

0

βk(X(s))ds)ζk

where the Yk are independent unit Poisson processes.

(Assume that there are only finitely many ζk.)


Diffusion limits??

Possible scaling limits of the form

Xn(t) = Xn(0) +1

n

∑k

Yk(n2

∫ t

0

βnk (Xn(s))ds)ζk +

∫ t

0

F n(Xn(s))ds

where Yk(u) = Yk(u)− u and F n(x) =∑

k nβnk (x)ζk.

Note that 1nYk(n

2·) ≈ Wk

Assuming Xn(0) → X(0), βnk → βk and F n → F , we might expect alimit satisfying

X(t) = X(0) +∑k

Wk(

∫ t

0

βk(X(s))ds)ζk +

∫ t

0

F (X(s))ds

Kurtz (1980), Ethier and Kurtz (1986), Section6.5.


Uniqueness question

X(t) = X(0) +∑k

Wk(

∫ t

0

βk(X(s))ds)ζk +

∫ t

0

F (X(s))ds

Let τk(t) =∫ t

0 βk(X(s))ds and γ(t) =∫ t

0 F (X(s))ds. Then

τl(t) = βl(X(0) +∑k

Wk(τk(t))ζk + γ(t))

γ(t) = F (X(0) +∑k

Wk(τk(t))ζk + γ(t))

Problem: Find conditions under which pathwise uniqueness holds.


Compatibility for multiple time-changes

X(t) = X(0) +m∑k=1

Wk(

∫ t

0

βk(X(s))ds)ζk +

∫ t

0

F (X(s))ds

Set τk(t) =∫ t

0 βk(X(s))ds, and for α ∈ [0,∞)m, define

FYα = σ(Wk(sk) : sk ≤ αk, k = 1, 2, . . .) ∨ σ(X(0))

and

FXα = σ(τ1(t) ≤ s1, τ2(t) ≤ s2, . . . : si ≤ αi, i = 1, 2, . . . , t ≥ 0).

If X is a compatible solution, then Wk(∫ t

0 βk(X(s))ds), k = 1, . . . ,mare martingales with respect to the same filtration and hence X is asolution of the martingale problems for

Af(x) =1

2

∑i,j

aij(x)∂i∂jf(x) + F (x) · ∇f(x),

a(x) =∑

k βk(x)ζkζTk .


Two-dimensional case

For i = 1, 2, Wi standard Brownian motions.

βi : R2 → (0,∞), bounded

X1(t) = W1(

∫ t

0

β1(X(s))ds) X2(t) = W2(

∫ t

0

β2(X(s))ds)

or equivalently

τi(t) = βi(W1(τ1(t)),W2(τ2(t))), i = 1, 2

A strong, compatible solution exists, and weak uniqueness holds byStroock-Varadhan, so pathwise uniqueness holds.


ReferencesH. 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 inProbability 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. ISSN0090-9491.

Rajeeva L. Karandikar. On pathwise stochastic integration. Stochastic Process. Appl., 57(1):11–18, 1995. ISSN0304-4149.

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.

http://dx.doi.org/10.1214/EJP.v12-431

http://dx.doi.org/10.1214/EJP.v12-431


AbstractTime-change equations for diffusion processes

General notions of weak and strong solutions of stochastic equations will be described and a general versionof the Yamada-Watanabe-Engelbert theorem relating existence and uniqueness of weak and strong solutionsgiven. Time-change equations for diffusion processes provide an interesting example. Such equations arisenaturally as limits of analogous equations for Markov chains. For one-dimensional diffusions they also areessentially given in the now-famous notebook of Doeblin. Although requiring nothing more than standardBrownian 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 ofcompatible solutions used to reduce the uniqueness question to the corresponding question for a martingaleproblem or an Ito equation.

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