Mean-field backward doubly stochastic differential equations and related SPDEs
School of Mathematics, Shandong University, Jinan, 250100, China
School of Mathematics, Shandong Polytechnic University, Jinan, 250353, China
Boundary Value Problems 2012, 2012:114 doi:10.1186/1687-2770-2012-114Published: 17 October 2012
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.