Averaging principles for forward-backward multivalued stochastic systems and applications to systems of nonlinear parabolic partial differential equations

This work concerns a type of stochastic systems in which the forward equations are general stochastic differential equations and the backward equations are stochastic variational inequalities. We first prove an averaging principle for general stochastic differential equations in the $L^{2p}$ ($p\geq 1$) sense. In addition, a convergence rate for $p=1$ is presented. Combining general stochastic differential equations with backward stochastic variational inequalities, we then establish another averaging principle for backward stochastic variational inequalities in the $L^{2}$ sense using a time discretization method. Finally, we apply our result to nonlinear parabolic partial differential equations to obtain their averaging principles.