Stochastic Recursive Optimal Control Problem with Mixed Delay under Viscosity Solution's Framework

TL;DR

This paper investigates stochastic recursive optimal control problems with mixed delay using viscosity solutions, establishing connections between Pontryagin's maximum principle and dynamic programming without derivatives of the value function.

Contribution

It introduces a viscosity solutions framework to relate adjoint processes and the value function in stochastic control with delays, without requiring derivatives.

Findings

Established relations between adjoint processes and the value function using viscosity solutions.

Provided a stochastic verification theorem for optimal control verification.

Discussed the connection between Pontryagin's maximum principle and Bellman's principle in delayed stochastic systems.

Abstract

This paper is concerned with the stochastic recursive optimal control problem with mixed delay. The connection between Pontryagin's maximum principle and Bellman's dynamic programming principle is discussed. Without containing any derivatives of the value function, relations among the adjoint processes and the value function are investigated by employing the notions of super- and sub-jets introduced in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is really optimal.