Yamada-Watanabe Results for Stochastic Differential Equations with Jumps

Abstract

Recently, Kurtz (2007, 2014) obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic equations covering also the case of stochastic differential equations with jumps. Following the original method of Yamada and Watanabe (1971), we give alternative proofs for the following two statements: pathwise uniqueness implies uniqueness in the sense of probability law, and weak existence together with pathwise uniqueness implies strong existence for stochastic differential equations with jumps.

1. Introduction

In order to prove existence and pathwise uniqueness of a strong solution for stochastic differential equations, it is an important issue to clarify the connections between weak and strong solutions. The first pioneering results are due to Yamada and Watanabe [1] for certain stochastic differential equations driven by Wiener processes.

We investigate stochastic differential equations with jumps. Let be a second-countable locally compact Hausdorff space equipped with its Borel -algebra . Let be a -finite Radon measure on , meaning that the measure of compact sets is always finite. Let be disjoint subsets. Let . Let , , , and be Borel measurable functions, where is equipped with its Borel -algebra (see, e.g., Dudley [2, Proposition ]). Consider a stochastic differential equation (SDE) where is an -dimensional standard Brownian motion, is a Poisson random measure on with intensity measure , , and is a suitable process with values in .

Yamada and Watanabe [1] proved that weak existence and pathwise uniqueness imply uniqueness in the sense of probability law and strong existence for the SDE (1) with and . Engelbert [3] and Cherny [4] extended this result to a somewhat more general class of equations and gave a converse in which the roles of existence and uniqueness are reversed; that is, joint uniqueness in the sense of probability law (see, Engelbert [3, Definition ]) and strong existence imply pathwise uniqueness. The original Yamada-Watanabe result arises naturally in the procedure of proving existence of solutions of a SDE; for a detailed discussion, see Kurtz [5, pages 1-2].

Jacod [6] generalized the above mentioned result of Yamada and Watanabe for a SDE driven by a semimartingale, where the coefficient may depend on the paths both of the solution and of the driving process. The Yamada-Watanabe result has been generalized by Ondreját [7] and Röckner et al. [8] for stochastic evolution equations in infinite dimensions and by Tappe [9] for semilinear stochastic partial differential equations with path-dependent coefficients.

Recently, there has been a renewed interest in generalizations of the results of Yamada and Watanabe [1]. Kurtz [5, 10] continued the direction of Engelbert [3] and Jacod [6]. He studied general stochastic models which relate stochastic inputs with stochastic outputs and obtained a general version of the Yamada-Watanabe and Engelbert theorems relating existence and uniqueness of weak and strong solutions of stochastic models with the message that the original results are not limited to SDEs driven by Wiener processes. In order to derive the original Yamada-Watanabe results from this general theory, proofs of pathwise uniqueness require appropriate adaptedness conditions, so two new notions, compatibility and partial compatibility between inputs and outputs, have been introduced. Due to Example 3.9 in Kurtz [10] and Page 7 in Kurtz [5], the results are valid for SDEs driven by a Wiener process and Poisson random measures.

Following the ideas of Yamada and Watanabe [1], we are going to give alternative proofs for the following two statements.

Theorem 1. Pathwise uniqueness for the SDE (1) implies uniqueness in the sense of probability law.

Theorem 2. Weak existence and pathwise uniqueness for the SDE (1) imply strong existence.

Note that Theorems 1 and 2 are generalizations of Proposition 1 and Corollary 1 in Yamada and Watanabe [1] (we do not intend to deal with generalization of their Corollary 3). The definition of weak and strong solutions of the SDE (1), pathwise uniqueness for the SDE (1) and uniqueness in the sense of probability law, and a detailed, precise formulation of Theorem 2 will be given in the paper. In the course of the proofs we developed a sequence of lemmas discussing several kinds of measurability; see Lemmas 12 and 14, and we also presented a key observation on the preservation of the joint distribution of the parts of the SDE (1); see Lemmas A.2 and A.4.

Our alternative proofs show the power of the original method of Yamada and Watanabe [1]; these proofs can be followed step by step and every technical detail is transparent in the paper. This raises a question whether Kurtz’s result could be proved via the walked-out path by Yamada and Watanabe.

Note that Situ [11, Theorem 137] also considered the SDE (1) with instead of and with and proved Theorems 1 and 2 under the resctrictive assumption This assumption was needed for introducing an auxiliary càdlàg process in Lemma 139 in Situ [11]. In fact, one can get rid of condition (2) by using the space of point measures on as the space of trajectories of Poisson point processes instead of the space of càdlàg functions; see the proofs of Theorems 1 and 2. We call the attention that in the literature the result of Situ [11, Theorem 137] has been usually referred to without checking condition (2); see, for example, Li and Mytnik [12, ], Dawson and Li [13, ], Döring and Barczy [14, ], and Li and Pu [15, and ], but Theorem 2 covers these situations as well.

We remark that Zhao [16] already adapted the original method of Yamada and Watanabe for the SDE (1) driven only by a compensated Poisson random measure, that is, with and , but for processes with values in a separable Hilbert space instead of -valued processes. Comparing with the results of the present paper, note that we explicitly stated and proved in Theorem 1 that pathwise uniqueness for the SDE (1) implies uniqueness in the sense of probability law.

2. Preliminaries

Let , , , , and denote the set of nonnegative integers, positive integers, real numbers, nonnegative real numbers, and positive real numbers, respectively. For , we will use the notation . By and , we denote the Euclidean norm of a vector and the induced matrix norm of a matrix , respectively. Throughout this paper, we make the conventions and for any with . By and , we denote the set of continuous and càdlàg -valued functions defined on , equipped with a metric inducing the local uniform topology (see, e.g., Jacod and Shiryaev [17, Section ]) and a metric inducing the so-called Skorokhod topology (see, e.g., Jacod and Shiryaev [17, Theorem ]), respectively. Moreover, and denote the corresponding Borel -algebras on them.

Recall that is a second-countable locally compact Hausdorff space. Note that is homeomorphic to a separable complete metric space; see, for example, Kechris [18, Theorem 5.3]. For our later purposes, we recall the notion of the space of point measures on , of the space of simple point measures on , and of the vague convergence. We follow Resnick [19, Chapter 3] and Ikeda and Watanabe [20, Chapter I, Sections 8 and 9].

A point measure on is a measure of the following form: let and let be a countable collection of (not necessarily distinct) points of , and let assuming also that for all and compact subsets (i.e., is a Radon measure meaning that the measure of compact sets is always finite, and consequently, it is locally finite), where denotes the Dirac measure concentrated on the point . Thus

A point function (or point pattern) on is a mapping , where the domain is a countable subset of such that is finite for all and compact subsets . The counting measure on corresponding to is defined by Note that there is a (natural) bijection between the set of point functions on and the set of point measures on with , , and . Namely, if is a point function, then the corresponding point measure is its counting measure . The set of all point measures on will be denoted by , and define a -algebra on it to be the smallest -algebra containing all sets of the form Alternatively, is the smallest -algebra making all the mappings , , , measurable.

Note that there is a (natural) bijection between the set of point processes (randomized point functions) defined on a probability space with values in the space of point functions on (in the sense of Ikeda and Watanabe [20, Chapter I, Definition 9.1]) and the set of -measurable mappings with for all and and for all (which are (special) point processes in the sense of Resnick [19, page 124]).

A point process on is called a Poisson point process if its counting measure is a Poisson random measure on (for the definition of Poisson random measure see, e.g., Ikeda and Watanabe [20, Chapter I, Definition 8.1]). A Poisson point process is stationary if and only if its intensity measure is of the form for some measure on , which is called its charateristic measure. If is a Radon measure, then is Poisson distributed with parameter ; hence is finite with probability one for all and compact subsets . Consequently, a stationary Poisson point process with a Radon charateristic measure is a stationary Poisson point process in the sense of Ikeda and Watanabe [20, Chapter I, Definition 9.1].

Next we recall vague convergence. Let be the space of -valued continuous functions defined on with compact support. For , , we say that converges vaguely to as if for all . For a topology on giving this notion of convergence, see page 140 in Resnick [19]. Recall that coincides with the Borel -algebra generated by the open sets with respect to the vague topology on ; see, for example, Resnick [19, Exercises and ].

In what follows we equip the spaces , , , and with some -algebras that will be used later on. For each , let us equip and with the -algebras for , respectively, where is the mapping which stops the function at . It is easy to check that, for all , coincides with the smallest -algebra containing all the finite-dimensional cylinder sets of the form and then see, for example, Problem in Karatzas and Shreve [21]. Similarly, for all , coincides with the smallest -algebra containing all the finite-dimensional cylinder sets of the form and then hence coincides with in Definition VI.1.1 in Jacod and Shiryaev [17]. Finally, let us equip with the -algebras , , being the smallest -algebra containing all sets of the form Note that since the union of the generator system of the -algebras , , forms a generator system of .

3. Notions of Weak and Strong Solutions

If is a probability space, then, by -null sets from a sub--algebra , we mean the elements of the set

Definition 3. Let be a probability measure on . A weak solution of the SDE (1) with initial distribution is a tuple , where (D1) is a filtered probability space satisfying the usual hypotheses (i.e., is right continuous and contains all the -null sets in );(D2) is an -dimensional standard -Brownian motion;(D3) is a stationary -Poisson point process on with characteristic measure ;(D4) is an -valued -adapted càdlàg process such that(a)the distribution of is ,(b), ,(c), ,(d), , where is the counting measure of on ,(e)equation (1) holds -a.s., where .

For the definitions of an -Brownian motion and an -Poisson point process, see, for example, Ikeda and Watanabe [20, Chapter I, Definition 7.2 and Chapter II, Definition 3.2].

In the next remark we point out that the integrals in the SDE (1) are well defined under the conditions of Definition 3 and have càdlàg modifications as functions of .

Remark 4. If conditions (D1), (D2), and (D4)(b) are satisfied, then is well defined and has continuous sample paths almost surely; see, Ikeda and Watanabe [20, Chapter II, Definition 1.9]. Indeed, is -adapted (since is -adapted and is measurable), is measurable (since is measurable, because it has right-continuous paths, see Karatzas and Shreve [21, Remark ], and is measurable), and , .
Concerning conditions (D4)(c) and (d), note that the mappings and are -predictable; see Lemma A.1.
Hence condition (D4)(c) is satisfied if and only if the mapping is in the (multidimensional version of the) class defined on page 62 in Ikeda and Watanabe [20], that is, if it is -predictable and there exists a sequence of -stopping times such that almost surely as and Indeed, if (D4)(c) holds then (17) is satisfied for for , where almost surely as . On the other hand, (17) implies for all and , and hence (D4)(c), because almost surely as .
Moreover, if conditions (D1), (D3), and (D4)(c) are satisfied, then the process is well defined and has càdlàg sample paths almost surely. Indeed, for each , see page 63 in Ikeda and Watanabe [20]. The integrand belongs to the (multidimensional version of the) class defined on page 62 in Ikeda and Watanabe [20]; hence the process on the right hand side is a square integrable -martingale; see page 63 in Ikeda and Watanabe [20]. By Theorem in Karatzas and Shreve [21], this process has a càdlàg modification. Here we point out that, for using this theorem, we need completeness and right continuity of the filtration . Further, we also obtain for all , since almost surely as .
Recalling that the mapping is -predictable, condition (D4)(d) is satisfied if and only if the mapping is in the (multidimensional version of the) class defined on page 61 in Ikeda and Watanabe [20].
Further, if conditions (D1), (D3), and (D4)(d) are satisfied, then, by definition, the process is well defined and has càdlàg sample paths, where is the domain of (being a countable subset of ). Indeed, for each , by definition, the mappings are right and left continuous, respectively.

Remark 5. If , then condition (D4)(d) is satisfied automatically, since then implies , and hence is a finite sum with probability one.

Remark 6. Note that if conditions (D1)–(D3) are satisfied, then and are automatically independent according to Theorem 6.3 in Chapter II of Ikeda and Watanabe [20], since the intensity measure of is deterministic.
Moreover, if is a weak solution of the SDE (1), then , , and are mutually independent, and hence , , and are mutually independent as well. Indeed, the conditional joint charateristic function of and the counting measure of with respect to equals to the product of the (unconditional) charateristic functions of and the counting measure of ; see (6.12) in Chapter II of Ikeda and Watanabe [20] applied with and , and then one can use Lemma in Karatzas and Shreve [21]. Since is measurable with respect to due to (D4), we have the mutual independence of , , and .
The thinnings and of onto and are again stationary -Poisson point processes on and , respectively, and their characteristic measures are the restrictions and of onto and , respectively (this can be checked calculating their conditional Laplace transforms; see Ikeda and Watanabe [20, page 44]).
Remark that for any weak solution of the SDE (1), , the Brownian motion and the stationary Poisson point processes and are mutually independent according again to Theorem 6.3 in Chapter II of Ikeda and Watanabe [20]. Indeed, one can argue as before taking into account also that the intensity measures of and are deterministic, and condition (6.11) of this theorem is satisfied, because and live on disjoint subsets of .

Definition 7. One says that pathwise uniqueness holds for the SDE (1) if whenever and are weak solutions of the SDE (1) such that , then .

Remark 8. One may also consider the following more strict definition of pathwise uniqueness. Namely, one could say that pathwise uniqueness holds for the SDE (1) if whenever and are weak solutions of the SDE (1) such that , then . Note that in this definition we require that is an -Brownian motion and an -Brownian motion as well, and since it is not necessarily true that is an -Brownian motion, it is not clear whether this more strict definition of pathwise uniqueness and the one given in Definition 7 are equivalent. According to Ikeda and Watanabe [20, Chapter IV, Remark 1.3], they are equivalent. We also point out that in our statements and proofs we use pathwise uniqueness in the sense of Definition 7, and we do not use the above mentioned equivalence of the two kinds of definitions.

Definition 9. One says that uniqueness in the sense of probability law holds for the SDE (1) if whenever and are weak solutions of the SDE (1) with the same initial distribution, that is, for all , then for all .

Now we define strong solutions. Consider the following objects:(E1)a probability space ;(E2)an -dimensional standard Brownian motion ;(E3)a stationary Poisson point process on with characteristic measure ;(E4)a random vector with values in , independent of and .

Remark 10. Note that if conditions (E1)–(E4) are satisfied, then , , and are automatically mutually independent according to Remark 6.

Provided that the objects (E1)–(E4) are given, let be the augmented filtration generated by , , and ; that is, for each , is the -field generated by and by the -null sets from (which is similar to the definition in Karatzas and Shreve [21, page 285]). One can check that(i) satisfies the usual hypotheses;(ii) is a standard -Brownian motion;(iii) is a stationary -Poisson point process on with characteristic measure . Indeed, by Remark 10, is a standard -Brownian motion, and is a stationary -Poisson point process on with characteristic measure . Hence, by Theorems 6.4 and 6.5 in Chapter II in Ikeda and Watanabe [20], has the strong Markov property with respect to the filtration . Then Proposition in Karatzas and Shreve [21] yields that the augmented filtration satisfies the usual hypotheses. Moreover, the augmentation of -fields does not disturb the definition of a standard Wiener process and a stationary Poisson point process; hence is a standard -Brownian motion, and is a stationary -Poisson point process on with characteristic measure . For the standard Wiener process, see, for example, Karatzas and Shreve [21, Theorem ]. The main point is to show that is independent of for all with , and is independent of for all with , detailed as follows (in order to shed some light what is going on behind). Let with , and . Then, by Problem in Karatzas and Shreve [21], there exists such that is a -null set from , where denotes the symmetric difference of and . Using that we get for all , where the last but one step follows from the independence of and . A similar argument shows the independence of and .

Definition 11. Suppose that the objects (E1)–(E4) are given. A strong solution of the SDE (1) on and with respect to the standard Brownian motion , the stationary Poisson point process and initial value , is an -valued -adapted càdlàg process with satisfying (D4)(b)–(d).

Clearly, if is a strong solution, then is a weak solution with initial distribution being the distribution of .

4. Proof of Theorem 1

Our presentation as follows is a generalization of the one given in Section in Karatzas and Shreve [21].

Let us consider a weak solution of the SDE (1) with initial distribution on . Then , . We put for , and we regard the solution as consisting of four parts: , , , and . Let us consider the product space equipped with the Borel -algebra see, for example, Dudley [2, Proposition ]. The quadruplet induce the probability measure on according to the prescription We denote by a generic element of . The marginal of on the -coordinate of is the probability measure on , the marginal on the -coordinate is an -dimensional Wiener measure on , the marginal on the -coordinate is the distribution on of a stationary Poisson point process on with characteristic measure . Moreover, the distribution of the triplet under is the product measure because is -measurable and , , and are independent; see Remark 6. Furthermore, .

The product space defined in (26) is a complete, separable metric space, since is a complete, separable metric space with the usual Euclidean metric, is a complete, separable metric space with a metric inducing the local uniform topology (see, e.g., Jacod and Shiryaev [17, Section ]), is a complete, separable metric space with a metric inducing the so-called Skorokhod topology (see, e.g., Jacod and Shiryaev [17, Theorem ]), and the vague topology on the space of all point measures on is metrizable as a complete, separable metric space (see, e.g., Resnick [19, Proposition 3.17, page 147]). Hence there exists a regular conditional probability for given , by an application of Karatzas and Shreve [21, Chapter 5, Theorem 3.19] with the random variable . We will be interested in conditional probabilities of sets in only of the form , where . Consequently, with a slight abuse of notation, there exists a function enjoying the following properties: (R1)for each , and , the set function is a probability measure on ;(R2)for each , the mapping is -measurable;(R3)for each and , we have We can call as the regular conditional probability for given .

Let us now consider two weak solutions , of the SDE (1) with the same initial distribution on ; thus According to (28), let and, as explained before, there exist functions enjoying the properties (R1)–(R3).

First, we bring the two triplets , , together on the same canonical space, while preserving the joint distribution of the coordinates within each triplet. Let equipped with the -algebra , which is the completion of the product -algebra by the collection of null sets under the probability measure for , where we have denoted by a generic element of , and then we extend to . Particularly, for all and , In order to endow with a filtration that satisfies the usual conditions, for each , we take , where the mapping is defined by and put We note that, for each , where , and the mapping is defined by for . Indeed, for all , by definition, the -algebra coincides with the -algebra generated by the sets for , , , , , and . Moreover, as in Problem in Karatzas and Shreve [21], the -algebra coincides with the -algebra generated by the sets for , , , , , , , , , and . Since, for any stochastic process , we get , .

The -coordinate process on induces a point process on with characteristic measure in a natural way, since, as it was recalled, there is a bijection between the set of point functions on and the set of point measures on with and , , and which follows from (34) using that is the distribution on of a stationary Poisson point process on with characteristic measure implying that

Next we check that , , are weak solutions of the SDE (1) with the same initial distribution . Using the definitions of , , , (R1) and (R3) we get for all and . Indeed, with , and , by Fubini theorem, So the distribution of under is the same as the distribution of under . Due to the definition of a weak solution, under , is an -dimensional standard -Brownian motion, and is a stationary -Poisson point process on with characteristic measure . Consequently, by the definition of (which is nothing else but the natural filtration corresponding to the coordinate processes), under , the -coordinate process is an -dimensional standard -Brownian motion, the process is a stationary -Poisson point process on with characteristic measure , and is -adapted, . Further, the same is true if we replace the filtration by ; see, Lemma A.5. Note also that the filtration satisfies the usual conditions. All in all, for each , the tuple satisfies (D1)–(D3).

Hence it remains to check that, for each , the tuple satisfies (D4). For each , let us apply Lemma A.4 with the following choices: Since is a weak solution of the SDE (1) with initial distribution , the tuple satisfies (D1)–(D4). Further, as it was explained before, the tuple satisfies (D1)–(D3), the process is adapted to the filtration , and the distribution of under is the same as the distribution of under . Then Lemma A.4 yields that the tuple satisfies (D4)(a)–(d) and the distribution of on under is the same as the distribution of on under , where is the counting measure of on , and . Using also that for each , the first process and the identically 0 process are indistinguishable (since the SDE (1) holds -a.s. for ), we obtain that the tuple satisfies (D4), as desired. It is worth mentioning that this is the place where we use that the filtration satisfies the usual conditions in order to ensure that the second process above has a càdlàg modification; see Remark 4. The filtrations and do not necessarily satisfy the usual conditions; this is the reason for introducing the filtration .

We have , because, by (45), , . Since , , are weak solutions of the SDE (1) with the same initial distribution , and , pathwise uniqueness implies , or equivalently, hence, applying (45), for all . Since , , and the mapping is continuous (see, e.g., Jacod and Shiryaev [17, Proposition ]), we have and then we obtain uniqueness in the sense of probability law.

5. Precise Formulation and Proof of Theorem 2

Our first result is a counterpart of Lemma 1.1 in Chapter IV in Ikeda and Watanabe [20] for stochastic differential equations with jumps, compare also with Situ [11, page 106, Fact A].

Lemma 12. If is a weak solution of the SDE (1) with initial distribution on , then for every fixed and , the mapping is -measurable, where denotes the completion of by the null sets of from .

Proof. Consider the regular conditional probability for given , where, for each , the stopped mapping is defined in (9), and , , ; that is, denotes the restriction of onto . The mapping enjoy properties analogous to (R1)–(R3). Namely, ()for each , , and , the set function is a probability measure on ;()for each , the mapping is -measurable;()for every and ,  where the probability measure is defined in (28). In order to prove the statement, it suffices to check that Indeed, then for and what is more, , since where and . Hence, for all , where Here , since, by , the set is in , and Further, and imply .
Unfortunately, (57) does not follow from the comparison of (R3) with , since still we do not know whether the function is -measurable. In order to show (57), it suffices to check that is valid for every . Indeed, then, by (R3), for all and , and hence, using also that the function is -measurable, by the uniqueness part of the Radon-Nikodým theorem, we have (57).
The class of sets satisfying is a Dynkin system; that is, (i), since and one can apply .(ii)If and , then . Indeed, (iii)If and , then . Indeed, by the continuity of probability and dominated convergence theorem, Consider the collection of sets of the form for , , and , where, for each , and are defined earlier, denotes the increment mapping , , , and denotes the increment mapping given by , , . This collection of sets is closed under pairwise intersection and generates the -algebra , since the collection of sets of the form with for , , , , and generates by (11), and the collection of sets of the form with for , , , and generates by (15). By the Dynkin system theorem (see, e.g., Karatzas and Shreve [21, Theorem ]), provided that we prove for of the form (69). For such a , by Fubini theorem, we have The fourth equality above follows from the -measurability of the function which is a consequence of and Fubini theorem. The fifth equality above follows from the independence of and under the measure ; see, for example, Ikeda and Watanabe [20, Chapter 2, Theorems 6.4 and 6.5]. For the last equality above we used and By (28), Therefore, if is of the form (69), then The second equality above follows from the independence of and under the probability measure . This independence holds because and is independent of under the probability measure ; see, for example, Ikeda and Watanabe [20, Chapter II, Theorems 6.4 and 6.5]. The relationship (76) is valid since , , and , the mapping is -measurable, and the mapping is -measurable, because the processes and are -adapted.

Remark 13. The filtration defined in Lemma 12 is the augmentated filtration generated by the coordinate processes on the canonical probability space . This is the counterpart of the augmentated filtration .

The next lemma is a generalization of Corollary 1 in Yamada and Watanabe [1] (see also Problem in Karatzas and Shreve [21]) for stochastic differential equations with jumps.

Lemma 14. Suppose that pathwise uniqueness holds for the SDE (1). If , , are two weak solutions of the SDE (1) with the same initial distribution on , then there exists a function such that holds for -almost every , where , , is given in (33). This function is -measurable, -measurable for every fixed , and

Proof. Fix and define the measure on the space equipped with the -algebra . By (34) and Fubini theorem, for all and . With the choice and , using that pathwise uniqueness holds for the SDE (1), relation (51) yields . Since for all , (79) yields the existence of a set with such that Again, by Fubini theorem, which can occur only if for some , call it , we have Indeed, since for all , , we have Since for all , by (R1), the set function is a probability measure on , , we get the unique existence of for all satisfying (82). Then we have (77) for .
For and any , we have if and only if , .
The aim of the following discussion is to show the -measurability of for all . For all and , we have where for . Lemma 12 implies , . Moreover, (due to the definition of , for more details, see the proof of Lemma 12); hence . Using that , and the definition of the augmented -algebra (see Lemma 12), we obtain . Hence , as desired.
The aim of the following discussion is to show that is where denotes the completion of with respect to the measure . For all , we have , where and are defined in (85). Property (R2) implies , . Moreover, by definition of completion (see, e.g., Definition in Karatzas and Shreve [21]), hence Using that , , by definition of completion, we obtain Hence as desired.
Next we check (78) for . For , by (45), (34), (R1), and (82), as desired.
It remains to check that one can choose a version of which is -measurable, -measurable for every fixed , and (77) and (78) remain hold for . Since is there exists a function which is -measurable and see, for example, Cohn [22, Proposition ]. First we check that is -measurable for every fixed . For all and , we have where (since is -measurable), (due to the definition of completion, since ), (since is a -algebra), and (due to the definition of completion, since ). Hence .
Next we check (77) for . Using that (77) holds for and , we have where for . This implies (77) for .
Finally, we check (78) for . First observe that , since, by (79), where we used (R1) as well. Then, by (45) and (34), for , we obtain where, for the last equality, we applied that (78) holds for .

Remark 15. Note that the function in Lemma 14 and the -null set on which (77) does not hold depend on the two weak solutions in question.

Applying Lemma 14 for weak solutions , , of the SDE (1) with the same initial distribution on , we obtain the following corollary.

Corollary 16. If pathwise uniqueness holds for the SDE (1) and is a weak solution of the SDE (1) with initial distribution on , then there exists a function such that holds for -almost every , where is given in (29). This function is -measurable, -measurable for every fixed , and .

Next we give the precise formulation of Theorem 2.

Theorem 17. Suppose that pathwise uniqueness holds for the SDE (1) and there exists a weak solution of the SDE (1) with initial distribution . Then there exists a function which is -measurable, -measurable for every fixed , and Moreover, if objects (E1)–(E4) are given such that the distribution of is , then the process is a strong solution of the SDE (1) with initial value .

Proof. Let for , , , where is as in Corollary 16. By Corollary 16, for the function , the desired measurability properties hold. Using Corollary 16 and , we have implying (99).
Note that, for , , and as described in (E1)–(E4), the triplets and induce the same probability measure on the measurable space with respect to the probability measure and , respectively, where denotes the probability measure appears in (E1), since , , and are -independent and , , and are -independent; see Remarks 6 and 10.
Observe also that the mappings are -measurable and respectively. Further, they are -measurable and for all , respectively. Indeed, since and are -measurable and -measurable, respectively, by (11) and (15), it is enough to check that for all , , , , , , , These relations hold since , , and , , are -measurable and -measurable, and and are -measurable and , -measurable, respectively. Similarly, one can argue that the functions in question are -measurable and -measurable for all , respectively.
Next, we check that the process is adapted to the augmented filtration . First, note that the process is adapted to if and only if is -measurable for all , where is given in (9). Indeed, where the last equivalence can be checked as follows. Since coincides with the smallest -algebra containing the finite-dimensional cylinder sets of the form it is enough to check that for all is equivalent with for all , , , , which readily follows from Since , , the mapping is -measurable for all , is -measurable for all , it remains to check that the mapping (104) is -measurable for all . Recall that where Since a generator system of together with is a generator system of , and we have already checked that the mapping (104) is it remains to verify that for all and . We show that for all , implying for all , as desired. If , then there exists such that and . Hence where, for the last but one equality, we used that the distribution of under is (as it was explained at the beginning of the proof). By definition, this means that .
Next we check that satisfies the SDE (1)  -almost surely. Since is -measurable, and the triplets and induce the same probability measure on the measurable space with respect to the probability measure and , respectively, the triplets and induce the same probability measure on the measurable space with respect to the probability measure and , respectively. Let us apply Lemma A.4 with the following choices: Since is a weak solution of the SDE (1) with initial distribution , the tuple satisfies (D1), (D2), (D3), and (D4)(b)–(e). Further, as it was explained before Definition 11, the tuple satisfies (D1), (D2), and (D3), and we have already checked that is adapted to the augmented filtration . Then Lemma A.4 yields that the tuple satisfies (D4)(b)–(d) and the distribution of on under is the same as the distribution of on under , where and is the counting measure of and on , respectively, and and . Using that the first process and the identically 0 process are indistinguishable (since the SDE (1) holds -a.s. for ), we obtain that the SDE (1) holds -a.s. for as well, that is, (D4)(e) holds.
Finally, we show that . Since, as it was checked that the distribution of and coincide, especially, the distribution of   and coincide, and consequently, the distribution of and coincide (both are equal to ). Using Corollary 16 for (which is especially a weak solution of the SDE (1) with initial distribution ) we get as desired.
Summarizing, is a strong solution of the SDE (1) with initial value .

Appendix

Let be a filtered probability space. First we recall the notion of -predictability; see, for example, Ikeda and Watanabe [20, Chapter II, Definition 3.3]. The predictable -algebra on is given by A function is called -predictable if it is -measurable.

Lemma A.1. Let be a filtered probability space. Let be an -adapted càdlàg process with values in .(i)If is a continuous function, then for each and , the function , , is -predictable.(ii)If , is an open set and , then (iii)If is -measurable, then the function , , is -predictable.

Proof. (i) The function is -predictable, since it belongs to the generator system of . Indeed, for each , the mapping is -measurable, because is -measurable and for all , and hence is -measurable, and is -measurable. Moreover, for each , the function is left continuous, because the functions and are left continuous and is continuous.
(ii) Consider the function given by , , where denotes the Euclidean distance of and . Then is continuous and . Put , . Then, by (i), we obtain (iii) We have , where , ; thus it suffices to show that is -measurable. The -algebra is generated by the sets with , open sets and ; hence it suffices to show that This holds by (ii).

Note that using Lemma A.1, one can relax Assumption in Applebaum [23].

The next lemma plays a similar role as Lemma 139 in Situ [11].

Lemma A.2. Let , , be tuples satisfying (D1), (D2), (D3), and (D4)(b)–(d). Suppose that and have the same distribution on . Then have the same distribution on , where, for each , is the counting measure of on , and .

Proof. By Remark 4, the above processes have càdlàg modifications. According to Lemma in Jacod and Shiryaev [17], it suffices to show that the finite-dimensional distributions of the above processes coincide.
By Proposition in Jacod and Shiryaev [17], for each and , and as , where Let , , be such that they are disjoint, , , and (such a sequence exists since is -finite; see, e.g., Cohn [22, page 9]). Then for each and , as   -almost surely, where where denotes the thinning of onto ; see, for example, Ikeda and Watanabe [20, page 62]. Since , by Remark 5, the set is finite -almost surely for all and , , and hence one can order the set according to magnitude, say ,   ,   . Namely, on the event having -probability 1, where we used that the point measure corresponding to the point function is its counting measure ; see Section 2. Then we can write in the form where is a finite sum -almost surely. Furthermore, by Remark 4, for and , as   -almost surely, where with for all , , satisfying -almost surely as . Let , , be such that they are disjoint, , , and (such a sequence exists since is -finite; see, e.g., Cohn [22, page 9]). Then, by pages 47 and 63 in Ikeda and Watanabe [20], for all , and , as , where By page 62 in Ikeda and Watanabe [20], for all , , , and , , where Similarly as for the integrals and , there exist sequences of random variables and such that and as , respectively. Then, for all and , as ; then , and, finally, . Using part (vi) of Theorem 2.7 in van der Vaart [24], we get for all , and , as . Since and have the same distribution, the random vectors have the same distribution for all , as well. Indeed, the random vectors above can be considered as some appropriate measurable function of and , respectively. For this, it is enough to verify that each coordinate of the above random vectors can be considered as some appropriate measurable function of and , respectively, hence we fix . (i)First observe that is a -measurable function of ; namely, , where is given by , .(ii)Next, is a -measurable function of as well; namely, , where is given by , .(iii)In a similar way, is a -measurable function of ; namely, , where is given by , , .(iv)Now we show that is a -measurable function of . As a first step, we show that for each there exist functions and such that is -measurable, is -measurable, and holds -almost surely. Then it will follow that , where given by  for is -measurable. To prove the existence of and , first we verify that is measurable with respect to the -algebra having the form  We have  for , , , . Indeed, on the one hand, if is such that and , then for each , there exists a unique with , and hence , and . On the other hand,  For the second inclusion, for each , let us choose such that  If is such that and , then there exists a unique with , and hence we have for , , and for . Since the set on right hand side of (A.21) is in the -algebra given in (A.20) and is a generator system of , we readily get that the random variable is measurable with respect to the -algebra given in (A.20). Let us apply Theorem in Dudley [2] with the following choices:(a), ,(b), , ,(c), , . Then there exist functions and such that is -measurable, is -measurable, and holds on . Since , we have   -almost surely, as desired. In what follows we provide an alternative argument for verifying that is an -measurable function of with the advantage that the measurable function in question shows up explicitly. We have , where and for . Further, for all . Consider the mappings , , defined by and , , . By Proposition in Jacod and Shiryaev [17], the mappings , , are continuous at each point such that for all . Moreover, we have , where the mappings , , are given by , . Observe that, for each , we have for all (since for all ); hence, it remains to check that the mappings , , are -measurable. This follows from for all , and , which is a consequence of the definition of .(v)Finally, we verify that is a -measurable function of . Based on the findings for and , it is enough to check that  where and can be defined similarly as and for all and , respectively (replacing in the definitions and by and , resp.). Note that  for , , , , . Similarly, as it was explained in case of , one can approximate by -measurable functions of , which yields (A.25). Hence we obtain the statement.

Remark A.3. In case of and , the statement of Lemma A.2 basically follows by Exercise (5.16) in Chapter in Revuz and Yor [25]; see also Lemma in von Weizsäcker and Winkler [26].

Next we formulate a corollary of Lemma A.2.

Lemma A.4. Let be a tuple satisfying (D1), (D2), (D3), and (D4)(b)–(d) and let be another tuple satisfying (D1), (D2), and (D3) such that is an -valued -adapted càdlàg process. Suppose that and have the same distribution on . Then (D4)(b)–(d) hold for the tuple as well, and the processes (A.5) and (A.6) have the same distribution on .

Proof. First we check that for all . Since is -measurable and and have the same law, the processes and have the same law as well. Since the mapping is continuous (see, e.g., Ethier and Kurtz [27, Chapter III, Section 11, Exercise 26], or Barczy et al. [28, Proof of Lemma ]), and consequently -measurable, the processes and have the same distribution with respect to and , respectively. Since for all , this yields for all , as desired.
Similarly, one can check that for all , and It remains to check that for , where is the counting measure of on . Recall that, in the proof of Lemma A.2, , , have been chosen such that they are disjoint, , , and . Further, the set is ordered according to magnitude as , , ; see (A.9). Then, for each and , as -almost surely, where where denotes the thinning of onto . Since and have the same distribution with respect to and , respectively, and have the same distribution with respect to and , respectively, for all and (which can be checked in the same way as in the proof of Lemma A.2 by replacing with ). Consequently, and have the same distribution with respect to and , respectively, for all . Since for , we have (A.28). All in all, the tuple satisfies (D4)(b)–(d), and then Lemma A.2 yields that the processes (A.5) and (A.6) have the same distribution on .

The next lemma corresponds to Fact B on page 107 in Situ [11].

Lemma A.5. Let us consider the filtered probability space given in the proof of Theorem 1. The process , , is an -dimensional standard -Brownian motion, and the process , , is a stationary -Poisson point process on with characteristic measure under the measure .

Proof. Using that the -coordinate process is an -dimensional standard -Brownian motion under , for the first statement, it is enough to prove the independence of and for every with . For this, it is sufficient to show Indeed, if , then there exists some such that , and consequently . Then, Moreover, if , then for all , and hence By dominated convergence theorem, using that has continuous sample paths -almost surely, we get that is, Thus, in the light of Lemma of Karatzas and Shreve [21], we get the independence of and for every with .
Using that is independent of under , we obtain for all and ; hence we conclude (A.31) and then the first statement.
Using that the process is a stationary -Poisson point process on with characteristic measure , as it was explained in the proof of the first statement, for the second statement, it is enough to show that for every with , every , every disjoint subsets and , Using that , , are independent of each other and from under , we get for all . The last but one equality above is a consequence that is a Poisson distributed random variable with parameter , , under . Hence we conclude the second statement as well.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of Mátyás Barczy and Gyula Pap was realized in the frames of TÁMOP 4.2.4. A/2-11-1-2012-0001 “National Excellence Program, Elaborating and operating an inland student and researcher personal support system.” The project was subsidized by the European Union and cofinanced by the European Social Fund. Zenghu Li has been partially supported by NSFC under Grant no. 11131003 and 973 Program under Grant no. 2011CB808001.