Relationship between stochastic maximum principle and dynamic programming principle under convex expectation

TL;DR

This paper explores the connection between the stochastic maximum principle and dynamic programming principle in forward-backward control systems under convex expectations, especially G-expectation, providing new insights into their relationship under different smoothness conditions.

Contribution

It establishes the relationship between MP and DPP under convex expectation with smooth value functions and introduces a method to handle non-smooth cases using first-order jets.

Findings

Derived the relationship between MP and DPP under smooth conditions.

Established the first-order sub-jet and super-jet for non-smooth value functions.

Provided estimates linking the value function and control system under G-expectation.

Abstract

In this paper, we study the relationship between maximum principle (MP) and dynamic programming principle (DPP) for forward-backward control system under consistent convex expectation dominated by G-expectation. Under the smooth assumptions for the value function, we get the relationship between MP and DPP under a reference probability by establishing a useful estimate. If the value function is not smooth, then we obtain the first-order sub-jet and super-jet of the value function at any t. However, the processing method in this case is much more difficult than that when t equals 0.