Connection between MP and DPP for Stochastic Recursive Optimal Control Problems: Viscosity Solution Framework in General Case

TL;DR

This paper explores the relationship between the stochastic maximum principle and dynamic programming for recursive control problems with non-convex control domains, establishing key set inclusions without smoothness assumptions.

Contribution

It establishes the connection between maximum principle and dynamic programming in a general setting, including non-convex control domains and non-smooth value functions.

Findings

Set inclusions among sub- and super-jets of the value function and adjoint processes

Comparison with classical results to derive adjoint equations

Extension of the viscosity solution framework to general stochastic control problems

Abstract

This paper deals with a stochastic recursive optimal control problem, where the diffusion coefficient depends on the control variable and the control domain is not necessarily convex. We focus on the connection between the general maximum principle and the dynamic programming principle for such control problem without the assumption that the value is smooth enough, the set inclusions among the sub- and super-jets of the value function and the first-order and second-order adjoint processes as well as the generalized Hamiltonian function are established. Moreover, by comparing these results with the classical ones in Yong and Zhou [{\em Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, 1999}], it is natural to obtain the first- and second-order adjoint equations of Hu [{\em Direct method on stochastic maximum principle for optimization with recursive utilities, arXiv:1507.03567v1 [math.OC], 13 Jul. 2015}].