Table of Contents
International Journal of Stochastic Analysis
Volume 2017, Article ID 2876961, 12 pages
https://doi.org/10.1155/2017/2876961
Research Article

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

Claremont Graduate University, Claremont, CA, USA

Correspondence should be addressed to Henry Schellhorn; ude.ugc@nrohllehcs.yrneh

Received 16 December 2016; Accepted 1 February 2017; Published 27 February 2017

Academic Editor: Lukasz Stettner

Copyright © 2017 Sixian Jin and Henry Schellhorn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.