Weak and strong solutions of general stochastic models

Typically, a stochastic model relates stochastic"inputs"and, perhaps, controls to stochastic"outputs". A general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations is given in this context. A notion of {\em compatibility} between inputs and outputs is critical in relating the general result to its classical forebears. The relationship between the compatibility condition and the usual formulation of stochastic differential equations driven by semimartingales is discussed.


Introduction and main theorem
This paper is essentially a rewrite of Kurtz (2007) following a realization that the general, abstract theorem in that paper was neither as abstract as it could be nor as general as it should be. The reader familiar with the earlier paper may not be pleased by the greater abstraction, but an example indicating the value of the greater generality will be given in Section 2. To simplify matters for the reader, proofs of several lemmas that originally appeared in the earlier paper are included, but the reader should refer to the earlier paper for more examples and additional references.
As with the results of the earlier paper, the main theorem given here generalizes the famous theorem of Yamada and Watanabe (1971) giving the relationship between weak and strong solutions of an Itô equation for a diffusion and their existence and uniqueness. A second reason for this rewrite is that the main observation that ensures that the main theorem gives the Yamada-Watanabe result is buried in a proof in the earlier paper. Here it is stated separately as Lemma 2.8.
The motivation of the original Yamada-Watanabe result arises naturally in the process of proving existence of solutions of a stochastic differential equation or, in the context of the present paper, existence of a stochastic model determined by constraints that may but need not be equations. The basic existence argument starts by identifying a sequence of approximations to the equation (or model) for which existence of solutions is simple to prove, proving relative compactness of the sequence of approximating solutions, and then verifying that any limit point is a solution of the original equation (model). The issue addressed by the Yamada-Watanabe theorem is that frequently, the kind of compactness verified is weak or distributional compactness. Consequently, what can be claimed about the limit is that there exists a probability space on which processes are defined that satisfy the original equation. Such solutions are called weak solutions, and their existence leaves open the question of whether there exists a solution on every probabiltiy space that supports the stochastic inputs of the model, that is, the Brownian motion and initial position in the original Itô equation context. The assertion of the Yamada-Watanabe theorem and Theorem 1.5 below is that if a strong enough form of uniqueness can be verified, then existence of a weak solution implies existence on every such probability space.
A stochastic model describes the relationship between stochastic inputs and stochastic outputs. For example, in the case of the Itô equation, b(X(s))ds, X(0) and W are the stochastic inputs and the solution X gives the outputs. Typically, the distribution of the inputs is specified (for example, the initial distribution is given and X(0) is assumed independent of the Brownian motion W ), and the model is determined by a set of constraints (possibly, but not necessarily, equations) that relate the inputs to the outputs. In the general setting here, the inputs will be given by a random variable Y with values in a complete, separable metric space S 2 and the outputs X will take values in a complete, separable metric space S 1 . For the Itô equation, we could take Let P(S 1 × S 2 ) be the space of probability measures on S 1 × S 2 , and for random variables (X, Y ) in S 1 × S 2 , let µ XY ∈ P(S 1 × S 2 ) denote their joint distribution. Our model is determined by specifying a distribution ν for the inputs Y and a set of constraints Γ relating X and Y . Let P ν (S 1 × S 2 ) be the set of µ ∈ P(S 1 × S 2 ) such that µ(S 1 × ·) = ν, and let S Γ,ν be the subset of P ν (S 1 × S 2 ) such that µ XY ∈ S Γ,ν implies (X, Y ) meets the constraints in Γ. Of course, since we are not placing any restriction on the nature of the constraints, S Γ,ν could be any subset of P ν (S 1 × S 2 ).
For a second example, consider a typical stochastic optimization problem.
Example 1.1 Suppose Γ 0 is a collection of constraints of the form where ψ ≥ 0 and |f i (x, y)| ≤ ψ.
In the terminology of Engelbert (1991) and Jacod (1980), µ ∈ S Γ,ν is a joint solution measure for our model (Γ, ν). Any distribution µ ∈ S Γ,ν determines a weak solution for our model (Γ, ν), that is, there exists a probability space on which are defined random variables (X, Y ) such that Y has distribution ν and (X, Y ) meet the constraints in Γ. We have the following definition for a strong solution.
If a strong solution exists on some probability space, then a strong solution exists for any Y with distribution ν. It is important to note that being a strong solution is a distributional property, that is, the joint distribution of (X, Y ) is determined by ν and F . The following lemma helps to clarify the difference between a strong solution and a weak solution that does not correspond to a strong solution.
c) µ corresponds to a strong solution if and only if η(y, dx) = δ F (y) (dx).
Proof. Statement (a) is a standard result on the disintegration of measures. A particularly nice construction that gives the desired G in Statement (b) can be found in Blackwell and Dubins (1983). Statement (c) is immediate.
We have the following notions of uniqueness.
Definition 1.4 Pointwise (pathwise for stochastic processes) uniqueness holds, if X 1 , X 2 , and Y defined on the same probability space with µ X 1 ,Y , µ X 2 ,Y ∈ S Γ,ν implies X 1 = X 2 a.s. Joint uniqueness in law (or weak joint uniqueness) holds, if S Γ,ν contains at most one measure.
Uniqueness in law (or weak uniqueness) holds if all µ ∈ S Γ,ν have the same marginal distribution on S 1 .
We have the following generalization of the theorems of Yamada and Watanabe (1971) and Engelbert (1991).
Theorem 1.5 The following are equivalent: a) S Γ,ν = ∅, and pointwise uniqueness holds. b) There exists a strong solution, and joint uniqueness in law holds.
Remark 1.6 In the special case that all constraints are given by simple equations, for example, then Proposition 2.10 of Kurtz (2007) shows that pointwise uniqueness, joint uniqueness in law, and uniqueness in law are equivalent. Note that stochastic differential equations are not of the form (1.1) (see Section 2), and the equivalence of uniqueness in law and joint uniqueness in law does not follow from this proposition in that setting; however, Cherny (2003) has shown the equivalence of uniqueness in law and joint uniqueness in law for Itô equations for diffusion processes.
The main result in Kurtz (2007), Theorem 3.14, was stated assuming the compatibility condition to be discussed in the next section and under the assumption that S Γ,ν was convex. Neither assumption is needed for Theorem 1.5. The compatibility condition is critical to showing that Theorem 1.5 implies the classical Yamada-Watanabe result as well as a variety of more recent results for other kinds of stochastic equations. (See Kurtz (2007) for references.) The convexity assumption is useful in giving the following additional result.
Corollary 1.7 Suppose S Γ,ν is nonempty and convex. Then every solution is a strong solution if and only if pointwise uniqueness holds.

Compatibility
It is not immediately obvious that Theorem 1.5 gives the classical Yamada-Watanabe theorem since proofs of pathwise uniqueness require approriate adaptedness conditions in order to compare two solutions. This leads us to introduce the notion of compatibility. In what follows, if S is a metric space, then B(S) will denote the Borel σ-algebra and B(S) will denote the space of bounded, Borel measurable functions; if M is a σ-algebra, B(M) will denote the space of bounded, M-measurable functions. Let E 1 and E 2 be complete, separable metric spaces, and let D E i [0, ∞), be the Skorohod space of cadlag E i -valued functions. Let Y be a process in D E 2 [0, ∞). By F Y t , we mean the completion of σ(Y (s), s ≤ t).
where {F X,Y t } denotes the complete filtration generated by (X, Y ) and {F Y t } denotes the complete filtration generated by Y .
This definition is essentially (4.5) of Jacod (1980). If Y has independent increments, then We will consider a more general notion of compatibility. If B S 1 α is a sub-σ-algebra of B(S 1 ) and X is an S 1 -valued random variable on a complete probability space (Ω, F , P ), Definition 2.2 Let A be an index set, and for each α ∈ A, let B S 1 α be a sub-σ-algebra of B(S 1 ) and B S 2 α be a sub-σ-algebra of B(S 2 ). The collection C ≡ {(B S 1 α , B S 2 α ) : α ∈ A} will be refered to as a compatibility structure.
Let Y be an S 2 -valued random variable. An S 1 -valued random variable X is C-compatible with Y if for each α ∈ A and each h ∈ B(S 2 ), Remark 2.4 In the temporally ordered setting, Buckdahn, Engelbert, and Rȃşcanu (2005) employ a similar martingale assumption.
Proof. Let {M α , α ∈ A} be a {F Y α }-martingale. For each α ∈ A, there must exist a Borel function h α such that M α = h α (Y ). Suppose α 1 ≺ α 2 . Then The proof of the converse is similar.
Note that (2.2) is equivalent to requiring that for each h ∈ B(S 2 ), so compatibility is a property of the joint distribution of (X, Y ). Consequently, compatibility is a constraint on joint distributions. To emphasize the special role of compatibility, S Γ,C,ν will denote the collection of joint distributions that satisfy the constraints in Γ and the C-compatibility constraint.
To prove pointwise (pathwise) uniqueness, we still need some way of comparing compatible solutions.
Definition 2.6 Let the random variables X 1 , X 2 , and Y be defined on the same probability space with X 1 and X 2 S 1 -valued and Y S 2 -valued.
Pointwise uniqueness for jointly C-compatible solutions holds if for every triple of processes (X 1 , X 2 , Y ) defined on the same probability space such that µ X 1 ,Y , µ X 2 ,Y ∈ S Γ,C,ν and (X 1 , X 2 ) is jointly compatible with Y , X 1 = X 2 a.s.
Uniqueness for jointly temporally compatible solutions is the usual kind of uniqueness considered for stochastic differential equations. The following lemma ensures that this kind of uniqueness is equivalent to the notion of pointwise uniqueness used in Theorem 1.5 and hence, for example, Theorem 1.5 implies the classical Yamada-Watanable theorem.
Lemma 2.7 Pointwise uniqueness for jointly C-compatible solutions in S Γ,C,ν is equivalent to pointwise uniqueness in S Γ,C,ν .
Clearly pointwise uniqueness in S Γ,C,ν implies pointwise uniquenes for jointly C-compatible solutions. The converse is a consequence of the following lemma.
In order to prove Lemma 2.8, we need the following technical lemma.
Lemma 2.9 X is C-compatible with Y if and only if for each α ∈ A and each g ∈ B(B S 1 α ), , and (2.4) follows. Conversely, for f ∈ B(S 2 ), g ∈ B(B S 1 α ), and h ∈ B(B S 2 α ), we have and compatibility follows.
Lemma 2.9 also gives the following result.
Proposition 2.10 If X is a strong, compatible solution, then F X α ⊂ F Y α for each α ∈ A. In particular, in the temporal compatibility setting, X is adapted to the filtration {F Y t }.
Consequently, g(X) is F Y α -measureable and hence F X α ⊂ F Y α .
Example 2.11 McKean-Vlasov limits lead naturally to stochastic differential equations of the form X(t) = X(0) + t 0 σ(X(s), µ X(s) )dW (s) + t 0 b(X(s), µ X(s) )ds where µ X(s) is required to be the distribution of X(s). Alexander Veretennikov raised the question of a Yamada-Watanabe type result for equations of this form. Setting Y = (X(0), W ) and requiring temporal compatibility, the set of joint solution measures S Γ,C,ν may not be convex. Consequently, the results of Kurtz (2007) may not apply. Theorem 1.5, however, does not assume convexity of S Γ,C,ν and consequently gives the desired result.