arXiv subjects

Detao Zhang

Publications and source records attributed to Detao Zhang.

2 recordsLinked to original sources

arXiv2022

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 (FBSDE). Within these families, one could get the same well-posedness of FBSDEs with totally different structures. The first family of FBSDEs are proved to be equivalent with respect to the Unified Approach. Thus one could get the well-posedness of the whole family if one member exists a unique solution. Another equivalent family of FBSDEs are investigated by introducing a linear transformation method. By reason of the fully coupling structure between the forward and backward equations, it leads to a highly interdependence in solutions. We are able to lower the coupling of FBSDEs, by virtue of the idea of transformation, without losing the well-posedness. Moreover, owing to the non-degeneracy of the transformation matrix, the solution to original FBSDE is totally determined by solutions of FBSDE after transformation. In addition, an example of optimal Linear Quadratic (LQ) problem is presented to illustrate.

math.OC↗

arXiv2015

On well-posedness of forward-backward SDEs-A unified approach

In this paper, we study the well-posedness of the Forward-Backward Stochastic Differential Equations (FBSDE) in a general non-Markovian framework. The main purpose is to find a unified scheme which combines all existing methodology in the literature, and to address some fundamental longstanding problems for non-Markovian FBSDEs. An important device is a decoupling random field that is regular (uniformly Lipschitz in its spatial variable). We show that the regulariy of such decoupling field is closely related to the bounded solution to an associated characteristic BSDE, a backward stochastic Riccati-type equation with superlinear growth in both components $Y$ and $Z$. We establish various sufficient conditions for the well-posedness of an ODE that dominates the characteristic BSDE, which leads to the existence of the desired regular decoupling random field, whence the solvability of the original FBSDE. A synthetic analysis of the solvability is given, as a "User's Guide," for a large class of FBSDEs that are not covered by the existing methods. Some of them have important implications in applications.

math.PR↗