Jiongmin Yong

Solving forward-backward stochastic differential equations explicitly - a four step scheme

SummaryIn this paper we investigate the nature of the adapted solutions to a class of forward-backward stochastic differential equations (SDEs for short) in which the forward equation is non-degenerate. We prove that in this case the adapted solution can always be sought in an “ordinary” sense over an arbitrarily prescribed time duration, via a direct “Four Step Scheme”. Using this scheme, we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic PDE system. Moreover the uniqueness of the adapted solutions (over an arbitrary time duration), as well as the continuous dependence of the solutions on the parameters, can all be proved within this unified framework. Some special cases are studied separately. In particular, we derive a new form of the integral representation of the Clark-Haussmann-Ocone type for functionals (or functions) of diffusions, in which the conditional expectation is no longer needed.

On semi-linear degenerate backward stochastic partial differential equations

Abstract. In this paper we study the well-posedness and regularity of the adapted solutions to a class of linear, degenerate backward stochastic partial differential equations (BSPDE, for short). We establish new a priori estimates for the adapted solutions to BSPDEs in a general setting, based on which the existence, uniqueness, and regularity of adapted solutions are obtained. Also, we prove some comparison theorems and discuss their possible applications in mathematical finance.

Stochastic Verification Theorems within the Framework of Viscosity Solutions

This paper studies controlled systems governed by Itos stochastic differential equations in which control variables are allowed to enter both drift and diffusion terms. A new verification theorem is derived within the framework of viscosity solutions without involving any derivatives of the value functions. This theorem is shown to have wider applicability than the restrictive classical verification theorems, which require the associated dynamic programming equations to have smooth solutions. Based on the new verification result, optimal stochastic feedback controls are obtained by maximizing the generalized Hamiltonians over both the control regions and the superdifferentials of the value functions.

Backward Stochastic Differential Equations

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).