Forward-Backward Stochastic Differential Equations Generated by Bernstein Diffusions

In this article, we prove new results regarding hitherto unknown relations that exist between certain Bernstein diffusions on the one hand and processes that typically occur in forward-backward systems of stochastic differential equations on the other hand. More specifically, we consider Bernstein diffusions that can wander in bounded convex domains where d is arbitrary, and which are generated there by a forward-backward system of decoupled linear deterministic parabolic partial differential equations. This makes them reversible Itô diffusions under some conditions that pertain to their marginal distributions, which then allows us to construct D × d ×d2 -valued processes that are weak solutions to suitably defined forward -backward systems of coupled stochastic differential equations. Moreover, we also consider the converse problem, namely, that of knowing whether the first component of a weak solution to a given forward-backward system is a Bernstein diffusion in some sense, which we solve affirmatively in a specific case.


Introduction and Outline
Bernstein diffusions constitute in some sense a generalization of Markov diffusions, and as such can be associated with certain classes of linear partial differential equations in a variety of ways. They are a particular case of the more general notion of reciprocal process that was thoroughly developed in [11] following the seminal contributions of [4] and [20], and have ever since played an important rôle in various areas of mathematical physics (see, e.g., [7,8,22,23] and references therein for a more complete account).
Under specific conditions regarding their marginal distributions, Bernstein diffusions are reversible and in that case exhibit a perfect symmetry between past and future. This comes about in a very natural way when they are associated with the Schrödinger equation of quantum mechanics as in [7], or with certain nonautonomous linear parabolic forward-backward boundary value problems as in [22]. In the latter case, the forward time and the backward time directions are introduced from the outset, so that the reversibility of the associated diffusions manifests itself by the fact that they satisfy both a forward and a backward Itô equation simultaneously, namely, one such equation for each time direction. This leads for instance to Feynman-Kac representations for the solutions of the given parabolic equations.
It is, therefore, natural to expect the existence of nontrivial connections between reversible Bernstein diffusions on the one hand, and the various processes that typically occur in forward-backward systems of stochastic differential equations on the other hand. Ever since the original works of [3,6,17] in the context of stochastic optimal control, the theory of such systems has indeed been considerably developed and generalized, which has led to many applications in areas such as option pricing problems in mathematical finance and partial differential equations, to name only two. In the latter case, this has come about mainly through various connections with viscosity solutions and Feynman-Kac representations for the solutions to nonlinear elliptic and parabolic problems (see, e.g., [9, 13-15, 18, 19], and their references). More recently (see [16] and the references therein), parts of that theory were extended to forward-backward systems that possess weak solutions in some sense, following many intermediary results on the subject which appeared since the publication of [2].
Our primary purpose in this article is, therefore, to bring about in a systematic way some relations between those two hitherto unrelated theories. Accordingly, we organize the remaining part of this article in the following way: In Section 2, we show how to generate reversible Itô diffusions from Bernstein processes associated with a forward-backward system of decoupled linear deterministic parabolic partial differential equations defined in an arbitrary bounded open convex subset of Euclidean space, thereby making reference to the theory developed in [22]. In Section 3, we use the results of Section 2 to define and construct a first type of forward-backward system of stochastic differential equations, using the property that the diffusions we just alluded to are weak solutions to certain forward Itô equations, whereas some functions of them lead to diffusions that are weak solutions to certain backward Itô equations. Our definition of a forward-backward system there is based on the fact that the Wiener processes which occur in our case are not given a priori, but rather determined a posteriori by the diffusions themselves, which makes it closely related to one of the notions of weak solution introduced in the third part of [2], or in the fifth part of [16]. In that way we generalize the traditional definitions found in the literature (see, e.g., [9] and [15]). In Section 3, we also investigate the converse question, namely, that of knowing whether the first component of a weak solution to a given forward-backward system is a Bernstein diffusion in some sense, answering this question affirmatively in a specific case. Finally, in Section 4, we define and construct a second kind of a forward-backward system, which is dual to that of Section 3. In that case indeed, we may consider the diffusions of Section 2 as weak solutions to certain backward Itô equations, whereas some appropriate functions of them lead to diffusions that satisfy forward Itô equations in a weak sense. Thereby, we show that the kind of duality we just alluded to is intimately tied up with the reversibility of the diffusions.
Throughout this article we use the standard notations for all the functional spaces we need without any further comments, referring the reader for instance to [1].

A Class of Reversible Itô Diffusions
Let D ⊂ R d be a bounded open convex subset whose smooth boundary ∂D is C 2+α with α ∈ (0, 1). We consider parabolic initial-boundary value problems of the form where T ∈ (0, +∞) is arbitrary and (., .) R d stands for the Euclidean inner product in R d . Furthermore, k is a matrix-valued function independent of x, l is an R d -valued vector-field while V and ϕ are real-valued functions, respectively. Moreover, the last relation in (1) stands for the conormal derivative of u with respect to the vector-field where n(x) denotes the unit outer normal vector at x ∈ ∂D, which we assume to be uniformly outward pointing, nowhere tangent to ∂D for every t ∈ [0, T ]. Whereas (1) is related to the forward time direction, its adjoint final-boundary value problem where ψ is a real-valued function as well, specifies the backward time direction. From this point of view we see that (1) and ( 3) constitute a forward-backward system of decoupled linear deterministic parabolic equations in D.
Regarding this system, we impose the following hypotheses: with k > 0 holds for all t ∈ [0, T ] and all q ∈ R d , and each component of the conormal vector-field satisfies (IF) We have ϕ, ψ ∈ C 2+α (D) with ϕ > 0 satisfying the conormal boundary condition relative to k at t = 0, and ψ > 0 satisfying that condition at t = T .
Let g and g * be the parabolic Green functions associated with (1) and (3), respectively. Under the above hypotheses, we have g * (y, s; x, t) = g(x, t; y, s) > 0 for all s, t ∈ [0, T ] with t > s, and furthermore the functions defined by and are classical positive solutions to (1) and (3), respectively. More specifically, we have where u ϕ is the unique classical positive solution to (1), and v ψ the unique classical positive solution to (3) (see, e.g., [10] for proofs of these standard properties is well defined and positive from (5) for all x, y, z ∈ D and all r, s, t satisfying r ∈ (s, t) ⊂ [0, T ] . Let us also consider the positive function μ given by for all E, F ∈ B(D), where we require that D×D dxdyϕ(x)g(y, T ; x, 0)ψ(y) = 1.
This means that μ defines a probability measure on B(D) × B(D), and it turns out that the knowledge of (9) and (10) is sufficient to imply the existence of a reversible Bernstein diffusion wandering in D and naturally associated with (1) and (3). To explain what this means and understand the significance of such an association in our context, we first recall what a Bernstein process is (see, e.g., [11] and the references therein for more general formulations): Definition 1. We say the D-valued process Z τ ∈[0,T ] defined on the complete probability space ( , F, P) is a Bernstein process if the following conditional expectations satisfy the relation for every bounded Borel mesurable function m : D → R, and for all r, Thus, a Bernstein process is a generalization of a Markov process in that its dynamical behavior at time r is solely determined by its behavior at times s and t. In other words, all past information gathered prior to time s is irrelevant, as is all future information accumulated after time t.
Bernstein processes are not reversible in general, but the fact is that the process Z τ ∈[0,T ] we associate with (1) and (3) is a reversible Itô diffusion, a fact that follows from the next result where we summarize those properties of Z τ ∈[0,T ] proved in [22], which are relevant in the sequel. Therein and further below we write F + τ ∈[0,T ] for the increasing filtration generated by the F + s 's, F − τ ∈[0,T ] for the decreasing filtration generated by the F − t 's and k 1 2 for the positive square root of the matrix k. (9) and (10) be valid. Then, there exist a probability space ( , F, P μ ) and a D-valued Bernstein process Z τ ∈[0,T ] on ( , F, P μ ) such that the following properties hold:

Proposition 1. Assume that Hypotheses (K), (L), (V), and (IF) hold, and let
for all E 0 , E T ∈ B(D). Furthermore, we have for every E ∈ B(D) and all r, such that the forward Itô equation holds P μ -a.s. for every t ∈ [0, T ], and there exists a d-dimensional Wiener process W τ ∈[0,T ] relative to the filtration F − τ ∈[0,T ] such that the backward Itô equation holds P μ -a.s. for every t ∈ [0, T ]. In (14) and (15), the forward and backward drifts are given by the vector fields respectively, where u ϕ and v ψ are given by (6)- (8).

Remarks.
(1) In the preceding statements and in the remaining part of this article, a stochastic integral of the form (14) is always meant to be a forward Itô integral defined with respect to F + τ ∈[0,T ] , while a stochastic integral of the form (15) is meant to be a backward Itô The fact that the process Z τ ∈[0,T ] remains confined to D is a consequence of the method of proof of Proposition 1 (see [22] for details). More generally, it is possible to construct Bernstein processes whose state space may be any a priori given σ -compact Hausdorff topological space endowed with its Borel σ -algebra (see, for instance, Theorem 2.1 in [11]). (3) Owing to the conormal boundary conditions in (1) and (3), the behavior of Z τ ∈[0,T ] within D is essentially that of a diffusion reflected at the boundary ∂D, as explained and illustrated in [22]. (4) It is plain from (12) that the function μ is the joint probability distribution of the process Z τ ∈[0,T ] , which satisfies in particular for all E, F ∈ B(D) according to (10). The initial marginal distribution density for Z 0 on the one hand, and the final marginal distribution density for Z T on the other hand, are then, respectively, given by and as a consequence of (5)- (7); they are thereby completely determined by the functions ϕ and ψ. Furthermore, the function P given by (9) is the transition function of Z τ ∈[0,T ] according to (13). (5) It is tempting to change the above approach around by rewriting the preceding two relations more suggestively as in terms of the single parabolic Green function g, and consider (20) as an inhomogeneous system of nonlinear integral equations in the unknowns ϕ and ψ with given right-hand sides μ 0 and μ T . The question is then whether (20) possesses a unique solution (ϕ, ψ). It follows from a very general theorem in [5] that the answer is affirmative as long as However, the main result in [5] we are referring to does not guarantee that ϕ and ψ are Hölder continuous even if μ 0 and μ T are, so that this version of things is not particularly useful in the context of this article. (6) Statement (b) of the proposition unveils what turns out to be an essential property of the process Z τ ∈[0,T ] in the next two sections, to wit, the fact that it may simultaneously and independently be considered as a forward and a backward Itô diffusion whose dynamics are governed by (14) and (15), respectively. This reversibility of Z τ ∈[0,T ] has its origins in the specific form (10) of the joint probability distribution μ.
In the next section, we define and construct a forward-backward system out of the fact that Z τ ∈[0,T ] satisfies (14) as a forward Itô diffusion. There we write k −1 for the inverse of the matrix k, which exists by virtue of (4).

A First Forward-Backward System of Stochastic Differential Equations
Let us begin by defining the translated and rescaled forward drift for all (x, t) ∈ D × [0, T ]. From the regularity of v ψ stated in the preceding section, we remark that each component of c * is only once continuously differentiable in x since (21) involves the gradient of v ψ . However, as we now want to investigate the dynamics of the stochastic process c * (Z τ , τ ) τ ∈[0,T ] , we ought to require that each component of c * be twice continuously differentiable in x according to Lemma 1 below. This, in turn, requires v ψ to be thrice continuously differentiable in x. Whereas this can be arranged by imposing more regularity conditions on ∂D, the coefficients and the final condition in (3), we refrain from writing those conditions out explicitly in order to avoid too many technicalities. We encode instead the required regularity of v ψ in the following hypothesis: We then have the following result: The hypotheses are the same as in Proposition 1. Then we have P μ -a.s. for every t ∈ [0, T ].
While (22) is just (14) rewritten in terms of ( 21), the proof of (23) requires the following preparatory result, showing that c * is a classical solution to a coupled system of backward semilinear parabolic partial differential equations. We omit the proof since it follows from (3), (21), and (R v ) through straightforward but lengthy calculations:

Lemma 1. Assume that all the hypotheses of Proposition 2 hold. Then, for each
We then have the following: Proof of Proposition 2. It remains to show that (23) holds. For this it is sufficient to prove that for each i ∈ {1, . . . , d} we have P μ -a.s. for all s, t ∈ [0, T ] satisfying t > s, for then (23) follows by replacing t by T and s by t in the preceding relation. From (22) and the usual forward Itô formula, we first obtain Furthermore, by using the partial differential equation in (24) and the fact that the matrix k(τ ) is symmetric, we can rewrite the first integrand on the right-hand side of the preceding expression as by virtue of the fact that The substitution of (27) into (26) then immediately leads to (25).
According to the outline of Section 1 and in light of the preceding result, the next step amounts to determining whether the stochastic processes Z τ ∈[0,T ] and c * (Z τ , τ ) τ ∈[0,T ] indeed satisfy a suitably defined forward-backward system of stochastic differential equations. What is required here is a definition that is tailored to fit the standard notion of such a system (see, e.g., [9] and [15]), up to the necessity of taking into account the fact that the Wiener process of Proposition 2 is not a priori given in contrast to the usual situations treated in those two and other references. The reason for this is that (22) and (23) are only satisfied in a weak sense, since the existence of the Wiener process W * τ ∈[0,T ] is subordinated to that of Z τ ∈[0,T ] . This motivates the following definition, where are two continuous functions and where the two forward Itô integrals are supposed to be well-defined. We write X = (X 1 , . . . , X d ) for any vector X ∈ R d 2 with each X i ∈ R d and |.| for the Euclidean norm in both R d and R d 2 : T ] defined on the complete probability space ( , F, P) is a solution to a forward-backward system of type I with initial condition η and final condition κ if the following four conditions hold: (a) The process Y τ ∈[0,T ] defined by is a continuous, square-integrable martingale on ( , F, P) relative to the increasing filtration F

(b) The process W t∈[0,T ] defined by
is a Wiener process with respect to F + τ ∈[0,T ] and we have

Remark.
A glance at the preceding notion shows that it is closely related to Definition 3.3 in Section 3 of [2], or to Definition 2.1 in [16]. We shall dwell a bit more on this further below when we examine the problem of uniqueness.
Let us now choose the functions in (28) as and for every i ∈ {1, . . . , d}, and let us write The preceding notation and the above considerations then lead to the following result:  Proof. According to Lemma 4 in [22] and to the very beginning of the proof of Theorem 3 in that article, the forward Wiener process W * t∈[0,T ] of Proposition 1 is given by with respect to the filtration F + τ ∈[0,T ] , where the continuous square-integrable martingale is In order to identify (30) with (22) and (31) with (23) when we take (32) and (33) into account, it is then sufficient to choose ] in Definition 2. In this case, (35) is (29), so that (30) indeed coincides with ( 22) when η = Z 0 by virtue of (21) and (32), this initial condition having the distribution density (18). Furthermore, (31) does identify with (23) when κ = c * (Z T , T ) because of (33), and the form (34) of the final condition is a consequence of (21) and the definition of ψ. The few remaining statements of the theorem follow from simple considerations.
The preceding existence result naturally raises the issue of uniqueness, as well as the converse question of knowing under what conditions the component A τ ∈[0,T ] of a solution (A τ , B τ , C τ ) τ ∈[0,T ] to a forward-backward system of type I with f and h as in the above theorem is in some sense identifiable with a Bernstein diffusion. It turns out that both problems are intimately related, as we now show.
Let θ * : D × [0, T ] → R d be a vector field such that θ * i ∈ C 2,1 (D × [0, T ]) for every i ∈ {1, . . . , d}. Let us define S θ * ,T as the set of all solutions to the forward-backward system of type I where f and h are given by (32) and (33), respectively, whose second component may be written as for every τ ∈ [0, T ]. We first have the following result:

Lemma 2. Every element of S θ * ,T is of the form
Furthermore, uniqueness in law holds in S θ * ,T .
Proof. The first statement follows from the usual identification procedure for this kind of agument, which amounts to using (30) and the forward Itô formula for (36) to obtain in particular for every t ∈ [0, T ] and every i ∈ {1, . . . , d}, after direct comparison with the last term on the right-hand side of (31). As for the second statement, for each j ∈ {1, 2} let (A j τ , θ * (A j τ , τ ), ∇θ * (A j τ , τ )) τ ∈[0,T ] ∈ S θ * ,T be defined on the complete probability space ( j , F j , P j ), and let us assume that the equality of the initial distributions holds for every E ∈ B(D). On the one hand, the preceding relation implies that for all E 0 , . . . , E n ∈ B(D) and all possible subdivisions 0 < τ 0 < τ 1 < ..... < τ n < T of [0, T ]. In order to see this we rewrite (30) as by virtue of (36). Since (32) with y = θ * (x, t) implies thatf is uniformly bounded in (x, t), and since k − 1 2 (t) is also uniformly bounded in t by virtue of (4), we have Consequently, (39) follows from (38) through an adaptation of classical considerations based on Girsanov's Theorem (see, e.g., Proposition 3.10 in Chapter 5 of [12] or the various considerations about uniqueness in [21]). On the other hand, (39) now implies by standard arguments that the processes (A The preceding lemma, thus, identifies a large class of solution-processes in which uniqueness in law holds, thereby providing a property of independent interest. In case θ * = c * , this observation and all the considerations of this section lead at once to the following result:

Theorem 2. Let
be defined on ( , F, P μ ), with Z τ ∈[0,T ] the forward Bernstein diffusion of Proposition 2 and for every E ∈ B(D). Then, processes (40) and (41) both have the same law; in particular, as a forward diffusion the process A τ ∈[0,T ] identifies in law with Z τ ∈[0,T ] . Expressed differently, the process (Z τ , c * (Z τ , τ ), ∇c * (Z τ , τ )) τ ∈[0,T ] of Theorem 1 is the unique solution in S c * ,T to the above forward-backward system of type I, where uniqueness is meant as uniqueness in law on ( , F, P μ ).

Remark.
It is interesting to compare the arguments of this section with those developed in the fifth part of [16], which are based on the validity of comparison principles for the viscosity solutions to certain quasilinear partial differential equations. To the best of our knowledge, however, and irrespective of the method used, the uniqueness question for forward-backward systems as general as those introduced in Definition 2 remains largely unresolved.
In the next section, we define and construct yet another forward-backward system, using this time the fact that Z τ ∈[0,T ] satisfies (15) as a backward Itô diffusion. For this we follow exactly the same pattern as above, while shortening or omitting some of the proofs since our arguments there are so to speak the symmetric versions of those developed thus far.

A Second Forward-Backward System of Stochastic Differential Equations
We begin by defining for all (x, t) ∈ D × [0, T ], and impose the following additional regularity condition for the same reasons as in the preceding section: We then proceed by investigating the dynamics of the stochastic process c(Z τ , τ ) τ ∈[0,T ] :

Proposition 3. The hypotheses are the same as in Proposition 1. Then we have
Proof. While (44) is just (15) rewritten in terms of (43), the components of c now satisfy the forward parabolic system , which follows from (1), (43), and (R u ). This allows us to prove that for each i ∈ {1, . . . , d}, P μ -a.s. for all s, t ∈ [0, T ] satisfying t > s. Indeed, from (44) and the backward Itô formula we obtain on the one hand, where we notice the important minus sign in front of the second term on the right-hand side. On the other hand, because of the partial differential equation in (46) we have which allows us to get (47) in exactly the same way as we obtained (25) in the proof of Proposition 2. Relation ( 45) then follows by setting s = 0 in (47).
The preceding result is thus dual to that of Proposition 2, since Z τ ∈[0,T ] is now a backward diffusion satisfying (44) while c(Z τ , τ ) τ ∈[0,T ] satisfies the forward equation (45) in a weak sense. This motivates the following definition, which, as in the preceding section, allows us to encode the preceding facts into a more general framework. The vector fields f and h are still as in (28) and the two backward Itô integrals there are supposed to be well defined: Definition 3. We say the D × R d × R d 2 -valued process (A τ , B τ , C τ ) τ ∈[0,T ] defined on the complete probability space ( , F, P) is a solution to a forward-backward system of type II with initial condition η and final condition κ if the following four conditions hold: (a) The process Y τ ∈[0,T ] defined by is a continuous, square-integrable martingale on ( , F, P) relative to the decreasing filtration F is a Wiener process with respect to F − τ ∈[0,T ] and we have

Choosing again
f (x, y, z, t) = l (x, t) + k(t)y, we obtain the following statement, dual to that of Theorem 1: The proof of Lemma 3 is identical to that of Lemma 2 and is thereby omitted. What the last statement of Lemma 3 means is that for j ∈ {1, 2} and A j τ , θ A j τ , τ , ∇θ A j τ , τ τ ∈[0,T ] ∈ S θ,T defined on the probability space ( j , F j , P j ), the equality P 1 κ 1 ∈ E = P 2 κ 2 ∈ E of the final distributions again leads to P 1 A 1 τ 0 ∈ E 0 , . . . , A 1 τ n ∈ E n = P 2 A 2 τ 0 ∈ E 0 , . . . , A 2 τ n ∈ E n so that the laws of the processes (A be defined on ( , F, P μ ), with Z τ ∈[0,T ] the backward Bernstein diffusion of Proposition 3 and for every E ∈ B(D). Then, processes (57) and (58) both have the same law; in particular, as a backward diffusion the process A τ ∈[0,T ] identifies in law with Z τ ∈[0,T ] . Expressed differently, the process (Z τ , c(Z τ , τ ), ∇c(Z τ , τ )) τ ∈[0,T ] of Theorem 3 is the unique solution in S c,T to the above forward-backward system of type II, where uniqueness is meant as uniqueness in law on ( , F, P μ ).

Concluding Remarks.
(1) The constructions of this article show that the reversible Bernstein diffusions associated with (1) and (3) are in a sense pivot processes which may be regarded as wandering off either to the future or to the past. In the first case they allow the generation of new processes that move into the past, thereby providing weak solutions to forward-backward stochastic systems of type I. In the second case they give rise to processes that move into the future, thereby generating weak solutions to forward-backward stochastic systems of type II. Finally, the first components of those weak solutions identify in law with reversible Bernstein diffusions. (2) We are not aware of any works that deal with forward-backward systems such as those introduced in Definitions 2 and 3, where the Wiener processes are not given a priori but only exist by virtue of the basic martingale properties of the other processes involved. This is presumably related to the fact that we need here two distinct filtrations to formulate the problems, in contrast to the usual approach where the Wiener process is given from the outset and where only one filtration is used. In any case, it would be interesting to get more general existence and uniqueness results for such systems that go beyond the specific cases investigated here.