Maximum principle for stochastic optimal control problem under convex expectation

TL;DR

This paper develops a maximum principle for stochastic optimal control problems under convex expectations, extending classical results to a broader expectation framework and applying it to linear quadratic control problems.

Contribution

It introduces a maximum principle under convex expectations dominated by G-expectation, including representation theorems and applications to linear quadratic control.

Findings

Established a variational equation for the cost functional.

Proved a sufficient maximum principle under convex assumptions.

Applied the maximum principle to linear quadratic control problems.

Abstract

In this paper, we study a stochastic optimal control problem under a type of consistent convex expectation dominated by G-expectation. By the separation theorem for convex sets, we get the representation theorems for this convex expectation and conditional convex expectation. Based on these results, we obtain the variational equation for cost functional by weak convergence and discretization methods. Furthermore, we establish the maximum principle which is sufficient under usual convex assumptions. Finally, we study the linear quadratic control problem by using the obtained maximum principle.