Infinite dimensional forward-backward stochastic differential equations and the KPZ equation

Sergio Almada Monter,Amarjit Budhiraja

doi:10.1214/ejp.v19-2709

Abstract

Kardar-Parisi-Zhang (KPZ) equation is a quasilinear stochastic partial differential equation (SPDE) driven by a space-time white noise. In recent years there have been several works directed towards giving a rigorous meaning to a solution of this equation. Bertini, Cancrini and Giacomin have proposed a notion of a solution through a limiting procedure and a certain renormalization of the nonlinearity. In this work we study connections between the KPZ equation and certain infinite dimensional forward-backward stochastic differential equations. Forward-backward equations with a finite dimensional noise have been studied extensively, mainly motivated by problems in mathematical finance. Equations considered here differ from the classical works in that, in addition to having an infinite dimensional driving noise, the associated SPDE involves a non-Lipschitz (specifically, a quadratic) function of the gradient. Existence and uniqueness of solutions of such infinite dimensional forward-backward equations is established and the terminal values of the solutions are then used to give a new probabilistic representation for the solution of the KPZ equation.

Highlights

In [16] probabilistic representations for solutions of certain quasilinear stochastic partial differential equations(SPDE) in terms of finite dimensional forward-backward stochastic differential equations have been studied
Backward stochastic differential equations have a long history of applications in financial mathematics; see [13] for a survey of the field; see [12] or [17] for some modern applications
In this work we give a probabilistic representation of the BCG solution of (1.1) through solutions of certain infinite dimensional forward-backward stochastic differential equations

Summary

Introduction

In [16] probabilistic representations for solutions of certain quasilinear stochastic partial differential equations(SPDE) in terms of finite dimensional forward-backward stochastic differential equations have been studied. In this work we give a probabilistic representation of the BCG solution of (1.1) through solutions of certain infinite dimensional forward-backward stochastic differential equations. The paper [3] shows that as k → ∞, hk converges in distribution (as a C([0, T ] : C(R)) valued random variable) to a limit process h, which is defined to be the solution of (1.1). Throughout this work, this process (strictly speaking – its probability law on C([0, T ] : C(R))) will be referred to as the BCG solution of the KPZ equation. A calculation similar to the one leading to (3.16) shows that U c is FS - adapted

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