Linear Forward - Backward Stochastic Di erential Equations

The problem of nding adapted solutions to systems of coupled linear forward-backward stochastic dierential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sucient condition for the solvability of the linear FBSDEs with the processes Z (serves as a correction, see x1) being absent in the drift. Then a Riccati type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step-Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati type equation is studied which leads to a representation of adapted solutions to linear FBSDEs. g t0 ; P) be a complete probability space on which dened a one dimensional standard Brownian motion W(t), such that fF t g t0 is the natural ltration generated by W(t), augmented by all the P-null sets in F. In this paper, we consider the following system of coupled linear forward-backward stochastic dierential equations author would like to thank Professor Z. Liu for his hospitality. Some stimulating discussions with Professor J. Ma of Purdue University deserves a special acknowledgment.