Non-Markovian Forward-Backward Stochastic Differential Equations with Discontinuous Coefficients

In this paper, the existence and uniqueness of a weak solution in the sense of [1] and [2] has been shown for a class of fully coupled forward-backward SDE (FBSDE) such that the forward drift coefficient is allowed to be discontinuous with respect to the backward component of the solution. The novelty of this paper lies on the fact that the FBSDE is non-Markovian, i.e., the coefficients of the FBSDEs are allowed to be random. This type of FBSDEs is inspired by the regime shift model, where the short term interest rate switches between regimes according to the rate level. As a consequence, the discontinuity of the system becomes inevitable, making it violate the usual assumptions of most existing results for FBSDEs. We show the weak well-posedness of the FBSDE by an approximation scheme, along with the decoupling strategy.


Introduction
The following forward-backward stochastic differential equations (FBSDE) on [ ] , t T are considered in this paper: where W is a standard Wiener process and coefficients b, σ , h and g are in general random. The coefficient b is allowed to have discontinuity in y. More . 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, σ , h and g are deterministic functions) has been discussed in [3]. For instance, the following is a Markovian-type "regime-switching" FBSDE: where the coefficients σ , h, and g are deterministic Lipschitz functions, but the drift coefficient b takes the following form: ∑   1 (4) where 1 2 a a −∞ < < < < ∞  is a finite partition of  , and i b '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

{ }
: 0 t r r t = ≥ be the short rate process, and ln t t X r = , 0 t ≥ . 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 θ 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. [4] [5]). 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 t s 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 [10]), and assuming the triggering level to be 0 α > we can derive a FBSDE with discontinuous coefficient: ≠ ; and , , α β σ are constants.
Clearly this is a special case of the FBSDE (3), and its strong solution under Markovian framework has been established in [3]. In this paper, we would like to extend the work of [3] and show the wellposedness of such FBSDE, namely, the existence and uniqueness of a solution ( ) (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.

Preliminary
Assumption 2.1. We assume the following standing assumption for this paper. It is well-understood that, in order to solve a fully coupled FBSDE one should look for a "decoupling random field" ( ) ,  -a.s. (cf. e.g., [11] , , We should note that a solution to the BSPDE is defined as the pair of progressively measurable random fields ( ) , u β . Clearly, when the coefficients are deterministic, we must have 0 β = 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 ( ) φ ∞ ∈   that satisfies the following conditions: e for large enough.
We shall call such a smooth function φ the weight function. One can easily check that if φ is a weight function, then one has Now for a given weight function φ , we denote 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 φ and the spaces in the notation, and denote simply as 0 H .
Clearly 0 H 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 H is a Hilbert space with the inner product 1 2 One can easily prove the integration by parts formula: for Similarly, we denote Furthermore, for any the following sense: for any It is worth noting that 1) For any two weight functions 1 2 , φ φ satisfying (9), there must exist con- 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 0 H .
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 ⊆   , In particular, if We now define the notion of Sobolev weak solutions to BSPDE (7). Definition 2.1. We say that the pair of random fields ( ) is uniformly bounded and, for any We say that ( ) u β is a weak solution such that Du is uniformly bounded.

A Priori Estimates and a Stability Result
H be a weak solution to the BSPDE (7). Then there exists a constant 0 C > , depending only on the bounds in Assumption 2.1, the duration T and the constant K φ for the given weight function (10), such that Proof. For simplicity, let us denote , , , f s x u γ as f. By integration by parts formula and a general Itô formula (see [13]), one has, There is only one troubled term in the above equality needs special treatment, where C is the generic constant described in the statement of the Proposition.
Thus (17) The proof can be completed by integrating the above inequality from s to T, and applying the Gronwall inequality.
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 x u v = . Hence 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, and this completes the proof.