A Dual Yamada--Watanabe Theorem for L\'evy driven stochastic differential equations

We prove a dual Yamada--Watanabe theorem for one-dimensional stochastic differential equations driven by quasi-left continuous semimartingales with independent increments. In particular, our result covers stochastic differential equations driven by (time-inhomogeneous) L\'evy processes.


Introduction
The classical Yamada-Watanabe theorem [18] for Brownian stochastic differential equations (SDEs) tells us that strong uniqueness (i.e. pathwise uniqueness) and weak existence implies weak uniqueness (i.e. uniqueness in law) and strong existence. Jacod [10] lifted this result to SDEs driven by semimartingales and extended it by showing that strong uniqueness and weak existence is equivalent to weak joint uniqueness and strong existence. For Brownian SDEs, results related to Jacod's theorem have also been shown by Engelbert [5].
It is not hard to see that for the SDE dX t = u t (X)dL t , L = Lévy process, u = 0, weak uniqueness implies weak joint uniqueness. Consequently, by Jacod's theorem, for such SDEs weak uniqueness and strong existence implies strong uniqueness and weak existence. This implication can be seen as a dual Yamada-Watanabe theorem. It is a natural and interesting question whether the dual theorem also holds for SDEs with possibly degenerate coefficients, i.e. when u = 0 is allowed. For SDEs driven by Brownian motion this question was answered affirmatively by Cherny [4]. More recently, the dual theorem has been generalized to various infinite-dimensional Brownian frameworks, see [15,16,17]. To our own surprise, a version for SDEs with Lévy drivers seems to be missing in the literature. Indeed, to the best of our knowledge, the only formulation of a dual theorem for SDEs with discontinuous noise is the main result of [19], which is a version for SDEs driven by time-inhomogeneous Poisson processes. However, the proof in [19] has a gap. 1 The purpose of this short paper is to close the gap in the literature and to prove a dual theorem for SDEs driven by quasi-left continuous semimartingales with independent increments (SIIs), which is a large class of drivers including in particular all Lévy processes. We now directly formulate our main result, where we refer to the next section for precise definitions. Let u and w be real-valued predictable processes on the path space of càdlàg functions R + → R and let L be an SII, which we parameterize below by its deterministic semimartingale characteristics.
the following are equivalent: (i) Strong uniqueness and weak existence holds.
(ii) Weak uniqueness and strong existence holds. (iii) Weak joint uniqueness and strong existence holds.
We shortly comment on the proof of Theorem 1. The implication (i) ⇒ (ii) is the classical Yamada-Watanabe theorem and the equivalence of (i) and (iii) is Jacod's theorem. Thus, it suffices to prove (ii) ⇒ (iii). More precisely, we prove that weak uniqueness implies weak joint uniqueness, see Theorem 3 below. Similar to Cherny's proof for Brownian SDEs, the main idea is to recover the driver L from the solution process X and an independent SII V . The technical core of the argument is the proof for independence of X and V . We first show that the covariation between X and V vanishes and then deduce independence from the weak uniqueness assumption with a martingale problem argument, which is inspired by ideas from [7]. Our proof for the independence of X and V is different from Cherny's argument, which uses a second auxiliary process. The proofs of the dual theorems in the recent works [15,16,19] are based on Cherny's argument, while in different settings, and the proof in [17] uses results from [16] and the so-called method of the moving frame.
Let us also mention an interesting follow up question: It is well-known that the Yamada-Watanabe theorem remains true for SDEs driven by a Poisson random measure, see [9,Theorem 14.94]. Thus, it is only natural to ask whether the dual theorem also holds for such SDEs. We leave this question for future research.
Finally, we end the introduction with a short comment on our notation: As far as possible, we use the notation developed in the monograph [11] of Jacod and Shiryaev. In particular, we use the Strasbourg notation for (stochastic) integrals, i.e. we write H • X for the stochastic integral of H with respect to X. Furthermore, we stress that throughout the paper equalities should be read up to a null set.
The article is structured as follows: In Section 2 we provide definitions and the statement that weak uniqueness implies weak joint uniqueness. The proof of this claim is given in Section 3.

The Setting and Main Results
In this section we introduce our setting in a precise manner. Let us start with a description of the random driver for the SDE (1.1). A càdlàg adapted process with initial value zero is called SII, if it is a quasi-left continuous semimartingale those semimartingale characteristics (B, C, ν) (relative to a fixed truncation function h : R → R) are deterministic. It is well-known ([11, Proposition II.2.9]) that the semimartingale characteristics (B, C, ν) of an SII admit a decomposition for an increasing continuous function A : R + → R + starting in the origin, dA t -integrable Borel functions b : R + → R and c : R + → R + , and a Borel transition kernel F from R + into R with F t ({0}) = 0 and · 0 (1 ∧ |y| 2 )F s (dy)dA s < ∞. We call such a quadruple (b, c, F ; A) local characteristic of the SII. Clearly, the local characteristics are not unique. Nevertheless, it is well-known ( [11,Theorem II.4.25]) that the law of an SII is fully characterized by its semimartingale characteristic and therefore also by its local characteristics. Equivalently, the class of SIIs can be defined as the class of quasi-left continuous semimartingales with independent increments, see [11,Theorem II.4.15]. This characterization explains the abbreviation SII which stands for semimartingale with independent increments. In view of this equivalent definition, it is evident that a Lévy process with (time-independent) Lévy-Khinchine triplet (b, c, F ) is an SII and (a version of) its local characteristics are given by (b, c, F ; A t = t).
Next, we introduce the coefficients of the SDE (1.1). Let D be the Skorokhod space, i.e. the space of càdlàg functions R + → R. We denote the coordinate process by X, i.e. X(ω) = ω for all ω ∈ D. Further, we equip D with the σ-field D σ(X t , t ∈ R + ) and the filtration (D t ) t≥0 , which is defined by D t s>t σ(X r , r ∈ [0, s]). Whenever we consider D, all terms such as predictable, local martingale, etc., refer to (D t ) t≥0 as underlying filtration.
Throughout the paper, we fix two real-valued predictable processes w and u on D, an initial value x 0 ∈ R and local characteristics (b, c, F ; A).
We are in the position to define existence and uniqueness concepts for the SDE (1.1). Hereby, we adapt terminology from [4,10].
is a filtered probability space with right-continuous and complete filtration which supports an SII L with local characteristics (b, c, F ; A).
Definition 2 ((Strong) Solution Process). Let (B, L) be a driving system. We call a càdlàg adapted process X on B a solution process to the SDE (1.1), if where it is implicit that the integrals are well-defined. The solution process X is called a strong solution process, if it is adaped to the completed natural filtration of L.
Definition 3 (Weak and Strong Existence). We say that weak (strong) existence holds for the SDE (1.1), if there exists a driving system which supports a (strong) solution process.
Definition 4 (Strong Uniqueness). We say that strong uniqueness holds for the SDE (1.1), if on any driving system there exists up to indistinguishability at most one solution process.
is called a (joint) solution measure for the SDE (1.1), if there exists a driving system (B, L) which supports a solution process X such that Q is the law of X (resp. the law of (X, L)).
Definition 6 (Weak (Joint) Uniqueness). We say that weak (joint) uniqueness holds for the SDE (1.1), if there exists at most one (joint) solution measure.
Theorem 2. For the SDE (1.1) the following are equivalent: (i) Strong uniqueness and weak existence holds.
(ii) Weak joint uniqueness and strong existence holds.
In other words, the equivalence of (i) and (iii) in Theorem 1 holds. In the next section we will prove the following: Theorem 3. For the SDE (1.1) weak uniqueness implies weak joint uniqueness.
Together with Theorem 2, this result implies Theorem 1.
Remark 1. [10,Theorem 7.2] shows that the existence of a strong solution measure even implies the existence of a strong solution process on every driving system. The definition of strong existence only requires the existence of one driving system supporting a strong solution process.
We end this section with some comments on applications. It is interesting to find analytic conditions implying strong uniqueness. For Brownian SDEs many such conditions are known, see, e.g., [6]. There are less results for SDEs with jumps, see, e.g., [2,14] for some recent recults. Due to Theorem 1, strong uniqueness can be deduced from strong existence together with weak uniqueness, which is for Lévy driven SDEs better understood than strong uniqueness (see, e.g., [13] for some recent results).
As a question for furture research, it would be interesting to find analytic conditions for strong existence without verifying directly strong uniqueness. We think that such conditions could lead, as a byproduct, to new results for strong uniqueness.
Let us already comment on a natural idea: One might think that weak convergence techniques could lead to strong existence, i.e. that linear growth and continuity conditions suffice for strong existence. Even when the coefficient is elliptic and weak uniqueness holds, this is not the case for SDEs driven by certain stable Lévy processes, see [1,3] and in particular the comment on p. 10 in [2].

Proof of Theorem 3
Let X be a solution process to the SDE (1.1) which is defined on a driving system ((Ω * , F * , (F * t ) t≥0 , P * ), L). Furthermore, let ((Ω o , F o , (F o t ) t≥0 , P o ), U ) be a second driving system, set and let F t be the P -completion of the σ-field s>t (F * s ⊗ F o s ). In the following B = (Ω, F, (F t ) t≥0 , P ) will be our underlying filtered probability space. We extend X, L and U to B by setting for (ω * , ω o ) ∈ Ω. Due to the results in [9, Section 10.2 b)], (B, L) and (B, U ) are driving systems and X is a solution process on (B, L). Next, we define a semimartingale V by The following lemma is proven after the proof of Theorem 3 is complete.
Lemma 1. The process V is an SII with local characteristics (b, c, F ; A). Moreover, V is independent of X. Because the distribution of (X, L) is completely determined by the distribution of (X, V ) and Lemma 1 yields that weak joint uniqueness is implied by weak uniqueness. The proof of Theorem 3 is complete.

Remark 2.
In case u is non-degenerate, i.e. u = 0, it is clear that V = U and consequently, Lemma 1 is trivial. In particular, for the key argument it is not necessary to introduce the auxilliary process U .
It remains to prove Lemma 1: Proof of Lemma 1: Step 1. Our first step is to show that V is an SII with local characteristics (b, c, F ; A). We compute its semimartingale characteristics (B V , C V , ν V ). Let us start with the following observation: Proof. It is clear that (U, L) is a two-dimensional semimartingale. Fix 0 ≤ s < t. By the construction of B and a monotone class argument, we see that

bounded and continuous and take
Thus, using that U and L have right-continuous paths, that for every closed set E ⊆ R 2 there exists a uniformly bounded sequence (f n ) n∈N of non-negative continuous functions such that f n (x) → 1 E (x), and a monotone class argument, we conclude that (U, L) has independent increments relative to the filtration (F t ) t≥0 . Finally, the structure of the characteristics follows from the independence of U and L, the Lévy-Khinchine formula, see [11,Theorem II.4.15], and [11,Lemma II.2.44]. Now, we are in the position to compute the characteristics of V . We start with the second characteristic C V . Note that and, by Lemma 2, that [U c , L c ] = 0. Thus, Next, we compute the third characteristic ν V . Using the formula for the third characteristic of (U, L) as given in Lemma 2, we obtain for every Borel function G : This shows that ν V (dt, dx) = F t (dx)dA t . Finally, we compute the first characteristic where we use that h ′ (0) = 0. The definition of the first semimartingale characteristic shows that Hence, we conclude that In summary, the process V is an SII with local characteristics (b, c, F ; A).
Step 2. We now show that V and X are independent. Hereby, we borrow ideas used in the proof of [7,Theorem 4.10.1]. For f ∈ C 2 b (R), which is the space of bounded twice continuously differentiable functions with bounded first and second derivative, we set for (x, t) ∈ R × R + . The next lemma is a martingale characterization for SIIs.
Lemma 3. Let Y be an adapted process with càdlàg paths which starts in the origin. Then, the following are equivalent: is a martingale.
Proof. Suppose that (i) holds and let f ∈ C 2 b (R) with inf x∈R f (x) > 0. Integration by parts yields that By [11,Theorem II.2.42], the process f (Y ) − · 0 Lf (Y s , s)dA s is a local martingale. Thus, also M f is a local martingale. Because h is a truncation function, there exists a constant ε > 0 such that h(x) = x whenever x ∈ (−ε, ε). Furthermore, by Taylor's theorem, for all y ∈ R Thus, t 0 (1∧|y| 2 )F s (dy)dA s < ∞ implies that the process M f is bounded on bounded time intervals and consequently, a martingale. We conclude that (ii) holds.
Next, we also give a martingale characterization for the set of solution measures to the SDE (1.1). For g ∈ C 2 b (R) and (ω, s) ∈ D × R + we set We have the following: if and only if Q(X 0 = x 0 ) = 1 and for all g ∈ C 2 b (R) the process Kg(X, s)dA s (3.5) is a local Q-martingale. Furthermore, for every solution process X to the SDE (1.1) the process K g • X is a local martingale on the corresponding driving system. Take f, g ∈ C 2 b (R) with inf x∈R f (x) > 0 and define M f and K g as in (3.3) and (3.5) with Y replaced by V and X replaced by X. As shown in Step 1, V is an SII with local characteristics (b, c, F ; A). Hence, M f is a martingale by Lemma 3. Similarly, because X is a solution process to the SDE (1.1), K g is a local martingale by Lemma 4. We now show that [M f , K g ] = 0. Recalling (3.4) and the proof of a) ⇒ c) in [11,Theorem II.2.42], we see that Furthermore, because ∆X = u(X)∆L, we have ∆K g = ∆g(X) = 0 if u(X) = 0 or ∆L = 0. Putting these pieces together, we conclude that s≤· ∆M f s ∆K g s = 0 and therefore For n ∈ N we define T n inf(t ∈ R + : |K g t | > n), K g,n K g ·∧Tn . Because K g has bounded jumps, K g,n is bounded on bounded time intervals. Thus, by integration by parts, [M f , K g,n ] = [M f , K g ] ·∧Tn = 0 implies that the process M f K g,n is a martingale which is bounded on bounded time intervals. Now, fix a bounded stopping time S and define a measure Q as follows: Clearly, as M f is a P -martingale starting in one, Q is a probability measure by the optional stopping theorem. Moreover, Q(X 0 = x 0 ) = 1. Because M f , K g,n and M f K g,n are P -martingales, we obtain for every bounded stopping time T that E Q K g,n T = E P M f S K g,n T = E P M f S 1 {S≤T } E P K g,n T |F S∧T + K g,n T 1 {T <S} E P M f S |F S∧T = E P M f S K g,n S∧T 1 {S≤T } + K g,n T M f S∧T 1 {T <S} = E P M f S∧T K g,n S∧T = 0. Thus, because T was arbitrary and T n ր ∞ as n → ∞, K g is a local Q-martingale. Furthermore, because g was arbitrary, using Lemma 4 and [10, Lemmata 2.7, 2.9] shows that the push-forward Q • X −1 is a solution measure to the SDE (1.1). Consequently, by the weak uniqueness assumption, P • X −1 = Q • X −1 .
Next, take F ∈ σ(X t , t ∈ R + ) such that P (F ) > 0 and set Using that P (F ) = Q(F ), we obtain that Thus, because S was arbitrary, M f is a Q * -martingale. Since f was arbitrary, we deduce from Lemma 3 that Q * • V −1 = P • V −1 and consequently, for every G ∈ σ(V t , t ∈ R + ) P (G, F ) = Q * (G)P (F ) = P (G)P (F ).
Because this equality holds trivially whenever P (F ) = 0, we conclude that V and X are independent. The proof is complete.