The maximum principle for optimal control of diffusions with non-smooth coefficients

Abstract

In this paper, we study the necessary conditions for optimality of a control problem where the state process is governed by a stochastic differential equation, with non differentiable coefficients. We obtain a stochastic maximum principle for this model. This is the first version of the stochastic maximum principle that covers the non smooth case. The proof is based on the approximation of the non smooth coefficients by smooth ones. Then we apply Ekeland's variational principle in order to establish some necessary conditions satisfied by ϵ-optimal controls. The result is obtained by passing at the limit by using Krylov's estimate. The adjoint process is defined in terms of almost everywhere derivatives of the coefficients, and is the unique solution of a linear backward stochastic differential equation