Maximum Principle of Stochastic Optimal Control Problems with Model Uncertainty
Tao Hao, Jiaqiang Wen, Jie Xiong
TL;DR
This paper develops a stochastic maximum principle for robust stochastic control in systems with Markovian regime switching and partial information, where model uncertainty affects both the state dynamics and the cost. It relies on forward-backward SDEs with regime switching, Bayes’ formula to rewrite the cost under a fixed probability measure, and weak convergence methods to obtain a variational inequality and the adjoint equations. The authors establish necessary and sufficient conditions for optimal controls, and illustrate the framework with a risk-minimizing portfolio problem under model uncertainty. The results advance robust control theory in regimes with both uncertain models and regime shifts, with direct implications for finance under misspecification and incomplete information.
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^β$-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 $θ$. 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.