Optimal stochastic investment games under Markov regime switching market
This paper focuses on stochastic investment games between two investors with incorporating the influence of the macro economical environment that modeled by a Markov chain with $d$ states. There are two correlated assets are available to two investors, each investor can only invest into one of assets and his opponent choose to invest the other one. The dynamic of the two assets are driven by two drifted Brownian motion with coefficients specified by the functions of the Markov chain. Thus the system considered in this paper is controlled SDEs with random coefficients. Only one payoff function is available to both investors, one investor wants to maximize the expected payoff function, while his opponent wants to minimize the quantity at the same time. As results, the existence of the saddle point of the game, a couple of equations satisfied by the value functions and optimal policies for both investors are derived. Based on finite-difference method and weak convergence theory, a vector-valued Markov chain is constructed for approximating the underlying risky process weakly, which enables us to obtain the value function and optimal policies numerically. To some extend, we can view this paper as a further research of the problems proposed in Wan [23].