Relationship between maximum principle and dynamic programming principle for recursive optimal control problem of stochastic evolution equations

TL;DR

This paper explores the connection between maximum principle and dynamic programming for stochastic evolution equations, addressing non-convex control domains and nonsmooth value functions, with theoretical and application insights.

Contribution

It establishes a link between MP and DPP using backward stochastic integral equations, and shows value function regularity under certain conditions.

Findings

Established relationship between first and second-order adjoint processes and value function derivatives.

Proved value function is $C^{1,1}$-regular under additional assumptions.

Presented applications demonstrating the theoretical results.

Abstract

This paper aims to study the relationship between the maximum principle and the dynamic programming principle for recursive optimal control problem of stochastic evolution equations, where the control domain is not necessarily convex and the value function may be nonsmooth. By making use of the notion of conditionally expected operator-valued backward stochastic integral equations, we establish a connection between the first and second-order adjoint processes in MP and the general derivatives of the value function. Under certain additional assumptions, the value function is shown to be $C^{1,1}$-regular. Furthermore, we discuss the smooth case and present several applications of our results.