On Stochastic Maximum Principle: A Backward Stochastic Partial\n Differential Equations Point of View

In this paper, we consider a class of stochastic control problems for\nstochastic differential equations with random coefficients. The control domain\nneed not to be convex but the control process is not allowed to enter in\ndiffusion term. Moreover, the terminal cost involves a non linear term of the\nexpected value of terminal state. Our purpose is to derive a new version of the\nPontryagin's stochastic maximum principle by adopting an idea inspired from the\nwork of Peng [S. Peng, Maximum Principle for Stochastic Optimal Control with\nNonconvex Control Domain, Lecture Notes in Control & Information Sciences, 114,\n(1990), pp. 724-732]. More specifically, we show that if we combine the spike\nperturbation of the optimal control combined with the stochastic Feynman-Kac\nrepresentation of linear backward stochastic partial differential equations\n(BSPDE, for short), a new version of the stochastic maximum principle can be\nderived. We also investigate sufficient conditions of optimality. In the last\npart of this paper, motivated by our version of SMP, an interesting class of\nforward backward stochastic partial differential equations is naturally\nintroduced and the solvability of such kind of equations is briefly presented.\n