Non-Zero-Sum Stochastic Differential Investment Games in Ambiguous Economy Based on CRRA Utility Criterion

With the development of society, the complexity of the model which needs to be solved is increasing, and the uncertainty of the model (also known as model ambiguity) is also expanding. In order to make the investment decisions more accurately with considering the model ambiguity, this paper studies the robust non-zero sum investment game between two competing ambiguity averse investors. Suppose that two investors can invest their wealth in a financial market composed of one risk-free asset and one risky asset, and use the relative performance to describe the competitive relationship between these two investors. A robust non-zero sum stochastic differential investment game model is constructed, and the HJB equation (Hamilton-Jacobi-Bellman equation) corresponding to the game problem is given by using the dynamic programming principle. The analytical expression of the equilibrium investment strategy and the corresponding value function are obtained by solving the HJB equation. The results show that: (1) considering model ambiguity can significantly increase the investor's utility compared with when model ambiguity is not considered; (2) The fierce market competition environment will cause herding among investors, imitate each other's investment decisions, and adopt risky investment strategies, thus increasing the systematic risk of the financial market; (3) compared with the traditional investment strategy (without considering the game), when considering the relative performance, investors under the Nash equilibrium strategy are more willing to take high risks to pursue high returns, thus widening the wealth gap between himself and his competitor; and the greater the sensitivity coefficient of investor's reaction (which can also reflect the intensity of market competition), the higher risk he prefer to take.