The following forward-backward stochastic differential equations (FBSDE) on are considered in this paper:
where W is a standard Wiener process and coefficients b, , h and g are in general random. The coefficient b is allowed to have discontinuity in y. More precisely, we assume
and . For simplicity, we shall assume that all the processes involved are 1-dimensional, but as we shall see later, in most of the situation the higher dimensional cases, especially for the forward component X and the Brownian motion W, can be argued in an identical way without substantial difficulties.
The FBSDE above is non-Markovian; namely, the coefficients are allowed to be random. The Markovian case (i.e., b, , h and g are deterministic functions) has been discussed in . For instance, the following is a Markovian-type “regime-switching” FBSDE:
where the coefficients , h, and g are deterministic Lipschitz functions, but the drift coefficient b takes the following form:
where is a finite partition of , and ’s are deterministic Lipschitz functions. The main feature of this FBSDE is that the coefficient b has, albeit finitely many, jumps in the variable Y. This type of FBSDEs is motivated by the following “regime-switching” term structure model that is often seen in practice. Consider, for example, the Black-Karasinski short rate model that is currently popular in the industry: let be the short rate process, and , . Then X satisfies the following SDE:
where W is a standard Brownian motion. A simple “regime-switching” version of (5) is that the mean reversion level shifts between two values . The switching in the short-rate is triggered by the level of the long rate. The existence of such structural shift was supported by empirical evidence (see, e.g.  ). Many dynamic models of the short rate have been proposed, and some of them are hidden Markovian in nature; that is, the switch is triggered by an exogenous factor (diffusion) process Y so that , where (see, e.g.,    ). As a result, “regime switch” models started to attract people’s attention. Sometimes business cycles are used as regime classification, while others might define regime by interest rate level. Either way, the important feature of regime switch model remains to be its accommodation to the interactions between regimes and dynamics of the interest rate. A typical approach to model regime switch incorporates a hidden Markov process as a state variable into the short rate dynamics. For example,
where is a Markov process with given transitional probability, is the noise. Introduction of the regime-dependence in these papers enriches the flexibility of the model and therefore leads to a higher capacity to fit empirical data.
In particular, if we consider the case in which the triggering process is the long term rate, then following the argument of a term structure model (see, for example, Duffie-Ma-Yong ), and assuming the triggering level to be we can derive a FBSDE with discontinuous coefficient:
where , is the long term treasury bond price; , ; and are constants.
Clearly this is a special case of the FBSDE (3), and its strong solution under Markovian framework has been established in . In this paper, we would like to extend the work of  and show the wellposedness of such FBSDE, namely, the existence and uniqueness of a solution to (2). The paper is organized as follows. In Section 2, we provide necessary preparations, establish assumptions and introduce notations. In section 3 we prove a priori estimates and a stability result. In Section 4 we prove a weak existence of the solution to the FBSDE (2). The main result of this paper is given in Section 5.
Assumption 2.1. We assume the following standing assumption for this paper.
(A.1) Each is bounded, continuous in s, and uniformly Lipschitz in x, with Lipschitz constant ;
(A.2) The function is continuous, and there exist constants such that . Furthermore, for fixed s, , along with its spatial derivatives, and are all uniformly Lipschitz in , with Lipschitz constant ;
(A.3) The function is bounded, continuous in s and uniformly Lipschitz in , with Lipschitz constant ;
(A.4) The function is bounded, smooth and Lipschitz in x, and , where the norm is define in (11).
It is well-understood that, in order to solve a fully coupled FBSDE one should look for a “decoupling random field” , such that , for all , -a.s. (cf. e.g., ). In the Non-Markovian case, the decoupling field should be a random field , and in light of the stochastic Feynman-Kac formula (cf. ), we expect that such a function u should solve the following quasilinear BSPDEs, in a certain sense:
We should note that a solution to the BSPDE is defined as the pair of progressively measurable random fields . Clearly, when the coefficients are deterministic, we must have and the BSPDE (7) is reduced to a quasilinear PDE.
We next introduce the notion of the weighted Sobolev space. We begin by considering a function that satisfies the following conditions:
We shall call such a smooth function the weight function. One can easily check that if is a weight function, then one has
Now for a given weight function , we denote to be the space of all Lebesgue measurable functions such that . When the weight function and the dimension of the domain and range spaces are clear from the context, and there is no danger of confusion, we often drop the subscript and the spaces in the notation, and denote simply as . Clearly is a Hilbert space equipped with the following inner product:
We can now define the Weighted Sobolev spaces as usual. For example, we shall denote to be the subspace of that consists of all those h such that its generalized derivative, still denoted as , is also in . Clearly, is a Hilbert space with the inner product , where . One can easily prove the integration by parts formula: for any and ,
Similarly, we denote to be the subspace of that contains all such that . Thus, is again a Hilbert space with inner product , where . Moreover, let be the dual space of , endowed with the dual product . Then is equipped with the following norm: . Clearly, in the sense that for any , it holds that
Furthermore, for any , in light of (12), we have in the following sense: for any , .
Remark 2.2. It is worth noting that
1) For any two weight functions satisfying (9), there must exist constants such that . So the norms defined via and are equivalent, and therefore, the spaces , , are independent of the choices of .
2) It is readily seen that the weight function belongs to the class of the so-called Schwartz functions, and consequently any functions with polynomial growth are in .
We conclude this section by introducing some spaces of stochastic processes that will be useful for the study of the BSPDEs. First, for any sub-σ-field , and , we denote to be the spaces of all -measurable, -integrable random variables. Next, for any generic Banach space , we denote to be all -valued, -progressively measurable random fields (or processes) such that
In particular, if , where is a given weight function, we denote , , respectively. Again, we often drop the subscript from the notations when the context is clear. Finally, the spaces of Banach-space-valued processes such as , for , are defined in the obvious way.
We now define the notion of Sobolev weak solutions to BSPDE (7).
Definition 2.1. We say that the pair of random fields is a weak solution to BSPDE (7) if is uniformly bounded and, for any , it holds that
We say that is a regular weak solution to BSPDE (7) if is a weak solution such that is uniformly bounded.
3. A Priori Estimates and a Stability Result
Proposition 3.1. Assume Assumption 2.1. Let be a weak solution to the BSPDE (7). Then there exists a constant , depending only on the bounds in Assumption 2.1, the duration T and the constant for the given weight function (10), such that
Proof. For simplicity, let us denote . Also let us denote as f. By integration by parts formula and a general Itô formula (see ), one has,
There is only one troubled term in the above equality needs special treatment, . For this purpose, we define Taking the derivative with respect to x to get . Hence
where C is the generic constant described in the statement of the Proposition. Thus (17) becomes
The proof can be completed by integrating the above inequality from s to T, and applying the Gronwall inequality.
Proposition 3.2. Let and , be a sequence of coefficients of BSPDE (7) satisfying Assumption 2.1 uniformly. Assume that
2) For any fixed , ;
3) For each l, BSPDE (7) with coefficients has a weak solution . The solutions are uniformly bounded in supremum norm and are uniformly bounded in , uniformly in l;
4) There exists such that .
Then and there exists such that is a weak solution to BSPDE (7) with coefficients .
Proof. By the definition of , and condition 3), are uniformly bounded in . So there exists such that weakly in . It is clear that v is bounded. Moreover, the differential operator with respect to x is a closed operator, that is, for any ,
This implies that . Hence and is bounded in . Next, denote . By Proposition 3.1, we know uniformly in l for some constant C. Thus we can extract a subsequence, still indexed by (l), such that
for some and any . Note that by the boundedness of and , we know that uniformly in l. We now define . Note that, for any ,
where the second convergence is due to (19) and the boundedness of , and the third convergence is due to the uniform boundedness of . That is, converges to weakly in . It remains to show that is a weak solution to BSPDE (7) with coefficients .
It suffices to check (15). We fix and a smooth function with compact support. For each l, since is a weak solution to the corresponding BSPDE, we have
Note that since the boundedness and the convergence properties of all the involved terms, and thanks to Assumption 2.1, there is only one term left to check, . To this end, we define and . Clearly and are continuous and converges to pointwise. Since are bounded, say, by a constant , we have
From this it is easy to see that converges to , and the limit is uniform in on any compact subset of . Using similar argument one can show that
Similar to Proposition 3.1, one gets
and this completes the proof.
4. Weak Existence
Parallel to the Itô-Krylov formula ( , Theorem 2.10.1), we provide a general version of Itô-Ventzell formula which requires weaker regularity conditions. The proof can be easily produced similarly to the proof of Itô-Krylov formula. The key point in the proof is the utilization of the boundedness and non-degeneracy assumptions of the coefficients. We refer the readers Theorem 2.10.1 of , and omit the proof.
Lemma 4.1. Consider the process
Let Q be a bounded region in and be the first exit time of the process from the region Q. Let be some Markov time such that . Assume further that there exist positive constants and , such that and for all and . Then for any Itô process F such that for -a.s. , and
where and , the following holds:
Now we are ready to show the weak existence of FBSDE (2). Let us first introduce some notations. For any , let be the partition on where , . For any , and , we denote
Let , where is the filtration generated by , and clearly for fixed ,
Similarly we define , , and . Clearly can be written in terms of in the canonical space. Let us denote
Then is a deterministic function from to . Similarly, we define , , and .
Proposition 4.2. Under Assumption 2.1, the FBSDE (2) admits a weak solution.
Proof. Step 1 (Weak Existence of the BSPDE): In Step 1 and Step 2, we assume that for all N, , , and are smooth functions except for the y component of . We consider the following FBSDE:
for any . Because of the assumption of smoothness, FBSDE (22) is a deterministic FBSDE, and by Theorem 5.1 in , FBSDE (22) admits a unique strong solution . Indeed, , where is the unique strong solution (again by Theorem 5.1 in ) of
Now let us define a function piecewisely, and this function is essential in finding the decoupling field of FBSDE (2). First for any and , define
Secondly, for each , we consider the FBSDE (22) on . By Theorem 5.1 in , for any , there exists a deterministic decoupling field , which is a weak solution to the corresponding PDE, such that
Furthermore, let us patch functions ’s together by defining the following function :
Thus on , we have
For any , denote and . Note that since all coefficients are bounded, we know that second derivatives of (see ). Here . By the Itô-Krylov formula, for and fixed x, one has
Thus we define
Again by the Itô-Krylov formula,
Note that , and . Thus (25) and (28) imply
Suppressing the variables , one has
and is a weak solution to BSPDE (30). By the regularity argument in , we know that are uniformly bounded in supremum norm on every compact subset of , and are uniformly bounded in . Thus we can extract a subsequence (still indexed by N) such that it converges to some . By Proposition 3.2, and there exists such that is a weak solution to BSPDE (7) with coefficients .
Step 2 (Weak Existence of the FBSDE): Now we use the decoupling field u to decouple FBSDE (2). Let us consider the following forward SDE:
Without loss of generality in what follows we assume . We claim that this SDE possesses a weak solution. Indeed, on any given probability space on which is defined a standard Brownian motion W, consider the following SDE:
Note that the function actually has a bounded spatial derivative . Combining with (A.1), it is readily seen that the coefficient is uniformly Lipschitz in x. Thus the SDE (32) admits a unique strong solution, denoted by .
Next, define , which is bounded, thanks to (A.1) and (A.2). Thus
is a martingale under . Now define , . Then by the Girsanov theorem ( , Theorem 8.9.4), under the process is a Brownian motion, and X satisfies (31). Let us abuse the notation and denote by W again. In other words, we have shown that there exist a process and Wiener process, still denoted by W, such that is a weak solution of (31). Let us define
We would like to show that is a weak solution of FBSDE (2). Since are bounded in , uniformly over N, we know for -a.s. . By the construction of and , it is also clear that and , where is defined in (8). Thus by Lemma 4.1, one has
which completes the proof of Step 2.
Step 3 (The General Case): Now let us consider the general case, that is, we proceed without requiring the smoothness of , , and . For each N let us consider the standard smooth mollifiers . Based on standard BSPDE result, it is easy to check that the conditions 1) - 4) in Proposition 3.2 hold. Following closely to Step 1 and Step 2, utilizing the stability result from Proposition 3.2, we conclude that FBSDE (2) admits a weak solution.
5. Weak Uniqueness
Now we are ready to prove the main result of this paper.
Theorem 5.1 Under Assumption 2.1, the FBSDE (2) admits a unique weak solution.
Proof. The weak existence has been shown in Proposition 4.2. Now suppose there is another weak solution of (2). Applying the general Itô-Ventzell formula to get
Note that h is Lipschitz continuous with Lipschitz constant K. Thus by Itô-Krylov formula again, one has
An application of the Gronwall inequality implies
Thus we shall consider the following decoupled forward system:
Step 3 (Forward Weak Uniqueness): From previous steps, clearly and are both weak solutions to (35).
Let , , and (note that the system is strictly non-degenerate). It is easy to see that
are martingales under probabilities P and , respectively. Define new probabilities and by
By the Girsanov theorem, and are Brownian motions under and , respectively. Since strong uniqueness holds for
and and are both weak solutions to the above equation, we know that the distribution of under coincides with the distribution of under , i.e., for any bounded measurable functional ,
Now for any , by (36),
Thus we have shown that the distribution of under P coincides with the distribution of under . By , we know that the distribution of under P coincides with the distribution of under .
Step 4 (Backward Weak Uniqueness): We know u is continuous, hence Lebesgue measurable. Note that
Hence it is easy to conclude that the law of under P coincides with the law of under .
The well-posedness in weak sense of a type of fully coupled FBSDE has been established. The difficulty of the problem stems from the fact that the forward drift coefficient may be discontinuous. The existence and uniqueness of a strong solution for such FBSDE remain to be an open problem. The author plans to attack this issue in his future research.
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