Stochastic McKean-Vlasov Control Problem with Regime-Switching and Its Applications

This paper focuses on the McKean-Vlasov system’s stochastic optimal control problem with Markov regime-switching. To this end, the authors establish a new Itô’s formula using the linear derivative on the Wasserstein space. This formula enables us to derive the Hamilton-Jacobi-Bellman equation and verification theorems for McKean-Vlasov optimal controls with regime-switching using dynamic programming. As concrete applications, the authors first study the McKean-Vlasov stochastic linear quadratic optimal control problem of the Markov regime-switching system, where all the coefficients can depend on the jump that switches among a finite number of states. Then, the authors represent the optimal control by four highly coupled Riccati equations. Besides, the authors revisit a continuous-time Markowitz mean-variance portfolio selection model (incomplete market) for a market consisting of one bank account and multiple stocks, in which the bank interest rate, the appreciation and volatility rates of the stocks are Markov-modulated. The mean-variance efficient portfolios can be derived explicitly in closed forms based on solutions of four Riccati equations.