Solutions Of Forward-backward Stochastic Differential Equations Research Articles

Two Equivalent Families of Linear Fully Coupled Forward Backward Stochastic Differential Equations

In this paper, we investigate two families of fully coupled linear Forward-Backward Stochastic Differential Equations (FBSDEs) and its applications to optimal Linear Quadratic (LQ) problems. Within these families, one could get same well-posedness of FBSDEs with totally different coefficients. A family of FBSDEs is proved to be equivalent with respect to the Unified Approach. Thus one could get well-posedness of whole family once a member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. Owing to the coupling structure between forward and backward equations, it leads to a highly interdependence in solutions. We are able to decouple FBSDEs into partial coupling, by virtue of linear transformation, without losing the existence and uniqueness to solutions. Moreover, owing to non-degeneracy of transformation matrix, the solution to original FBSDEs is totally determined by solutions of FBSDEs after transformation. In addition, an example of optimal LQ problem is presented to illustrate.

Adapted solution of a degenerate backward spde, with applications

In this paper we prove the existence and uniqueness, as well as the regularity, of the adapted solution to a class of degenerate linear backward stochastic partial differential equations (BSPDE) of parabolic type. We apply the results to a class of forward-backward stochastic differential equations (FBSDE) with random coefficients, and establish in a special case some explicit formulas among the solutions of FBSDEs and BSPDEs, including those involving Malliavin calculus. These relations lead to an adapted version of stochastic Feynman-Kac formula, as well as a stochastic Black-Scholes formula in mathematical finance.