Zero-sum Stochastic Differential Portfolio Games

Portfolio selection is one of the three pillars in mathematical finance. The importance of strategic decision making in uncertain circumstance is increasingly recognized in theory and practice. As an important instrument in mainstream economics, the methods and thoughts of games have been widely applied in optimal portfolio theory. There are many uncertain events in real world, such as war, disaster, inflation, and so on, which may influence the return rate and the volatility of the asset greatly. We investigate portfolio selection problem under more realistic asset price model by stochastic differential games theory.In this paper we investigate stochastic differential portfolio games in a simplified financial market consisting of a bank account and two risky assets. First, we study stochastic differential portfolio games in the case that the two investors can only invest in one of the risky assets and the bank account, under Vasicek stochastic interest model and Heston stochastic volatility model respectively. By invoking the using of the stochastic linear-quadratic control approach, we obtain the competitively-optimal strategies of the two investors and the optimal value functions of the games for a power utility function and a logarithmic utility function respectively. Then we study the stochastic differential portfolio games in the case that the two investors can invest in both of the two risky assets and the bank account. By using stochastic linear-quadratic control approach, we obtain the closed-form solutions to the game problems. Numerical examples are also provided to illustrate how the optimal portfolio strategies change when some model parameters vary.