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

TL;DR

This paper develops a new stochastic maximum principle for control problems with nonconvex control domains and non-linear terminal costs, using backward stochastic PDEs and spike perturbations.

Contribution

It introduces a novel version of the stochastic maximum principle leveraging backward SPDEs and spike perturbations for nonconvex control domains.

Findings

Derived a new stochastic maximum principle for complex control problems.

Connected the maximum principle with backward stochastic PDEs.

Explored solvability of a new class of forward-backward SPDEs.

Abstract

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