Maximum Principle of Stochastic Optimal Control Problems with Model Uncertainty

TL;DR

This paper develops a stochastic maximum principle for control problems with uncertain coefficients and regime switching, providing theoretical conditions and an application to portfolio risk minimization.

Contribution

It introduces a maximum principle framework for stochastic control with model uncertainty and regime switching, including new solutions for forward-backward SDEs and variational inequalities.

Findings

Established $L^eta$-solutions for FBSDEs with regime switching

Derived necessary stochastic maximum principle under uncertainty

Applied results to a risk-minimizing portfolio problem

Abstract

This paper is concerned with the maximum principle of stochastic optimal control problems, where the coefficients of the state equation and the cost functional are uncertain, and the system is generally under Markovian regime switching. Firstly, the $ L^\beta$-solutions of forward-backward stochastic differential equations with regime switching are given. Secondly, we obtain the variational inequality by making use of the continuity of solutions to variational equations with respect to the uncertainty parameter $\theta$. Thirdly, utilizing the linearization and weak convergence techniques, we prove the necessary stochastic maximum principle and provide sufficient conditions for the stochastic optimal control. Finally, as an application, a risk-minimizing portfolio selection problem is studied.