Viscosity Solutions for Mean Field Optimal Switching with a Two-Time-Scale Markov Chain

Abstract

In this paper, we consider the mean field optimal switching problem with a Markov chain under viscosity solution notion. Based on the conditional distribution of the Markov chain, the value function and corresponding dynamic programming principle (DPP) are established. We prove that the value function is the unique viscosity solution of the variational inequality on Wasserstein space. In particular, we consider a two-time-scale Markov chain and derive the convergence of the limit system. As an application of theoretical results, an innovative example concerning the stock trading problem in a regime switching market is solved.

Keywords: Viscosity solution, mean field optimal switching, variational inequality, Markov chain, two-time-scale