A global maximum principle for stochastic optimal control problems with delay and applications

Abstract

In this paper, an open problem is solved, for the stochastic optimal control problem with delay where the control domain is nonconvex and the diffusion term contains both control and its delayed term. Inspired by previous results about delayed stochastic control systems, Peng’s global stochastic maximum principle is generalized to the time delayed case. A special backward stochastic differential equation is introduced to deal with the cross terms, when applying the duality technique. Comparing with the classical result, the maximum condition contains an indicator function, which in fact is the characteristic of the stochastic optimal control problem with delay. Furthermore, to illustrate the applications of our theoretical results, three dynamic optimization problems are addressed based on the global maximum principle.