Backward Stochastic Differential Equations

Abstract

In Chapter 3, in order to derive the stochastic maximum principle as a set of necessary conditions for optimal controls, we encountered the problem of finding adapted solutions to the adjoint equations. Those are terminal value problems of (linear) stochastic differential equations involving the Ito stochastic integral. We call them backward stochastic differential equations (BSDEs, for short). For an ordinary differential equation (ODE, for short), under the usual Lipschitz condition, both the initial value and the terminal value problems are well-posed. As a matter of fact, for an ODE, the terminal value problem on [0, T] is equivalent to an initial value problem on [0, T] under the time-reversing transformation t T — t. However, things are fundamentally different (and difficult) for BSDEs when we are looking for a solution that is adapted to the given filtration. Practically, one knows only about what has happened in the past, but cannot foretell what is going to happen in the future. Mathematically, it means that we would like to keep the context within the framework of the Ito-type stochastic calculus (and do not want to involve the so-called anticipative integral). As a result, one cannot simply reverse the time to get a solution for a terminal value problem of SDE, as it would destroy the adaptiveness. Therefore, the first issue one should address in the stochastic case is how to correctly formulate a terminal value problem for stochastic differential equations (SDEs, for short).