Backward Stochastic Differential Equations and Applications

Abstract

A new type of stochastic differential equation, called the backward stochastic differentil equation (BSDE), where the value of the solution is prescribed at the final (rather than the initial) point of the time interval, but the solution is nevertheless required to be at each time a function of the past of the underlying Brownian motion, has been introduced recently, independently by Peng and the author in [16], and by Duffie and Epstein in [7]. This class of equations is a natural nonlinear extension of linear equations that appear both as the equation for the adjoint process in the maximum principle for optimal stochastic control (see [2]), and as a basic model for asset pricing in financial mathematics. It was soon after discovered (see [22], [17]) that those BSDEs provide probabilistic formulas for solutions of certain semilinear partial differential equations (PDEs), which generalize the well-known Feynmann-Kac formula for second order linear PDEs. This provides a new additional tool for analyzing solutions of certain PDEs, for instance reaction-diffusion equations.