The maximum principle for stochastic control problem with Markov chain in progressive structure

Abstract

In this paper, we obtain the stochastic maximum principle for progressive optimal control of the model driven by Brownian motions and Markov chain. The maximum principle in progressive structure is essentially different from that in the classical case. Our maximum principle has two parts, including continuous part and jump part, which are used to characterize the characteristics of control at continuous time and jump time respectively. This shows that the progressive structure can better describe the optimal control of stochastic system with jumps. Moreover, we give an LQ example to show that the cost functional can indeed get a smaller value in progressive structure. • Necessary and sufficient maximum principle in progressive structure. • Two maximum principle includes continuous part and jump part. • Progressive structure is essentially different from predictable structure. • Progressive structure accurately describes the feature of control for system with jump. • Progressive structure can find better control than predictable structure.