Maximum principle for an optimal control problem associated to a stochastic variational inequality with delay

Abstract

We deal with a stochastic control problem subject to a stochastic variational inequality with delay. By deriving the adjoint equation as an anticipated backward stochastic differential equation, we are able to establish necessary conditions of optimality under the form of a Pontryagin-Bensoussan stochastic maximum principle. This is achieved first for cadlag controls, by explicitly writing the coefficients of the adjoint equation in terms of the local time of the state process. The general result is then obtained by approximating the optimal control with continuous controls and applying Ekeland's variational principle to the approximating sequence.