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Open Access Research

Mean-field backward doubly stochastic differential equations and related SPDEs

Ruimin Xu

Author Affiliations

School of Mathematics, Shandong University, Jinan, 250100, China

School of Mathematics, Shandong Polytechnic University, Jinan, 250353, China

Boundary Value Problems 2012, 2012:114  doi:10.1186/1687-2770-2012-114

Published: 17 October 2012

Abstract

Existence and uniqueness result of the solutions to mean-field backward doubly stochastic differential equations (BDSDEs in short) with locally monotone coefficients as well as the comparison theorem for these equations are established. As a preliminary step, the existence and uniqueness result for the solutions of mean-field BDSDEs with globally monotone coefficients is also established. Furthermore, we give the probabilistic representation of the solutions for a class of stochastic partial differential equations by virtue of mean-field BDSDEs, which can be viewed as the stochastic Feynman-Kac formula for SPDEs of mean-field type.

Keywords:
mean-field; backward doubly stochastic differential equations; locally monotone coefficients; comparison theorem; stochastic partial differential equations