A stochastic maximum principle for mixed regular-singular control problems via Malliavin calculus

Our main concern in this paper is to study mixed regular-singular control problems, where the control variable has two components, the first being absolutely continuous and the second singular. The coefficients of the state process, as well as the running and final costs are random functions, so as the state process is no longer a Markov process. Our main result is to derive necessary conditions for optimality, also known as the Pontriagin stochastic maximum principle by using Malliavin calculus techniques. The adjoint process, which plays a key role in the stochastic maximum principle, is given by means of the Malliavin derivatives of the optimal state process.