Semigroup Solution of Path-Dependent Second-Order Parabolic Partial Differential Equations

Sixian Jin,Henry Schellhorn

doi:10.1155/2017/2876961

Abstract

We apply a new series representation of martingales, developed by Malliavin calculus, to characterize the solution of the second-order path-dependent partial differential equations (PDEs) of parabolic type. For instance, we show that the generator of the semigroup characterizing the solution of the path-dependent heat equation is equal to one-half times the second-order Malliavin derivative evaluated along the frozen path.

Highlights

In this paper we consider semilinear second-order pathdependent partial differential equations (PDEs) (PPDEs) of parabolic type
We first consider the heat equation expressed in terms of a backward time variable
By the functional Feynman-Kac formula introduced in [1, 6], it is immediate that 1/2ωt ∘ D2. is the generator of the semigroup of the pathdependent PDEs (PPDEs)

Summary

Introduction

In this paper we consider semilinear second-order pathdependent PDEs (PPDEs) of parabolic type. These equations were first introduced by Dupire [1] and Cont and Fournie [2] and will be defined properly . It turns out that 1/2Dxx, that is, one-half times the secondorder vertical derivative, is not the appropriate infinitesimal generator, because of path dependence. Consider a general path-dependent terminal condition Ψ(B), in [5], Jin et al gave a new representation of Brownian martingales Mt (with t ≤ T) as an exponential of a timedependent generator, applied to the terminal value MT ≡ Ψ(B): Mt. By the functional Feynman-Kac formula introduced in [1, 6], it is immediate that 1/2ωt ∘ D2. In the second part we characterize the generator of the semilinear PPDE

Martingale Representation

Representation of Solutions of PathDependent Partial Differential Equations