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. (C) 1997 Elsevier Science B.V.