WEAK EXISTENCE AND UNIQUENESS FOR FORWARD-BACKWARD SDEs

The authors of the present paper study the weak existence and the uniqueness in law for the solutions of a class of fully coupled forward-backward stochastic differential equations (FBSDEs) composed of the forward SDE dXt = σ(t, Xt, Yt)dWt + b(t, Xt, Yt, Zt)dt, X_0 = x, with non degenerate diffusion coefficient a = σσ∗ and driven by a d-dimensional Brownian motion W , and of a1-dimensional BSDE dYt =−f(t,Xt,Yt,Zt)dt+Ztσ(t,Xt,Yt)dWt, t∈[0,T], Y_T =G(X_T). The coefficients a and G are assumed to be globally Ho ̈lder continuous in x, uniformly with respect to t, y, while a and b are Lipschitz in (y, z), uniformly with respect to t, x. Finally, the driver f , being allowed to be of quadratic growth in z, is assumed to be locally Lipschitz in (y, z). To prove the weak existence of solutions the authors consider the associated PDE and adapt the skeleton of the so-called 4-step scheme introduced by J. Ma, P. E. Protter and J. M. Yong [Probab. Theory Re- lated Fields 98 (1994), no. 3, 339–359; MR1262970 (94m:60118)]: In a first step they construct a solution u of the associated PDE by using a regularization procedure. Then, by considering the martingale problem associated with the coefficients a(t, x, u(t, x)), b(t, x, (u, ∇xu)(t, x)) the authors construct a weak solution of the above FBSDE. This approach can be regarded as a decoupling of the forward and the backward SDEs. While the forward SDE is solved with the help of the martingale problem in the weak sense, the BSDE associates with the weak solution (Ω,F,P,FW,X,W,X) of the forward equation a strong, FW,X-adapted solution (Y,Z). The existence of such a strong solution is guaranteed by the assumptions on the driver f; the quadratic growth of f in z is handled with M. Kobylanski’s method [Ann. Probab. 28 (2000), no. 2, 558–602; MR1782267 (2001h:60110)]. The difficulties in the authors’ approach consist mainly in estimating the derivatives of u; this is done by applying the Caldero ́n-Zygmund theory. Finally, it has to be pointed out that the authors also prove the uniqueness in law of the weak solution.