Nonzero-sum Stochastic Differential Portfolio Games

ABSTRACT:Game theory is the study of rational decision making in an interactive environment, which is widely used in the area such as economics, finance, management, psychology, politics and so on. It has great theoretical and practical significance to study the portfolio selection problem between the investors under the uncertain environment by game theory.In this paper we consider the nonzero-sum continuous time stochastic differential investment portfolio game between two investors. A continuous-time financial market with three investment assets, namely, a bank account and two risky stocks, is considered. There are two investors in the financial market. While we allow both investors to invest freely in the bank account, one of the investor may trade only in the first risky stock, and similarly, the other investor may trade only in the second risky stock. The games considered here are nonzero-sum, so there are two utility functions which depend on both investors’ wealth processes. One investor chooses a dynamic portfolio strategy in order to maximize his terminal expected utility depending on the sum between their wealth processes, while the other investor acts antagonistically by choosing a dynamic portfolio strategy so as to maximize the other utility function. Firstly, suppose that the price processes of the risky stock satisfy the geometric Brownian motions. By invoking the use of the completion of square method, we obtain closed-form solutions to the nonzero-sum game problem for the exponential utility function. Numerical examples are also provided to illustrate how the optimal portfolio strategies change when some model parameters vary. Secondly, suppose that the price processes of the risky stock satisfy Markov-modulated geometric Brownian motion. We get the explicit expressions of the optimal portfolio strategies and the optimal value functions for the exponential utility functions by applying the dynamical programming principle (HJB equations).