N-agent and Mean Field Game for Optimal Investment and Risk Control Strategies

There are a large number of institutional investors in the financial market. The characteristics of institutional investors pursuing high returns and high wealth lead to increasingly fierce market competition. The competitive market environment makes institutional investors not only pursue the maximization of their own wealth, but also pay attention to the wealth gap between them and their competitors. The investment and risk control problem under the interaction of multiple institutional investors' strategies was studied, considering the situation that there are a large number of institutional investors in the market. It is assumed that each investor can invest his wealth in the financial market to realize wealth appreciation, and at the same time transfer the risks he faces to other financial institutions in the market by purchasing insurance and other means. The relative performance described by the difference between the investor's own wealth and the market average wealth is used to describe the market competition. The objective of the investor is to maximize the expected utility of the relative performance at the terminal moment. A multi person investment and risk control game model is constructed under the non-zero sum game framework. Taking the CARA utility function as an example, the optimal investment and risk control strategy under the Nash equilibrium state is obtained by using the stochastic differential game theory and the mean field game theory, and the sensitivity analysis of parameters is carried out. The research finds that: (1) Competition will lead to the rise of venture capital and the weakening of risk control, which will lead to the increase of systemic risk in the financial market; (2) The risk preference and market competition of institutional investors and their competitors will affect the balanced investment and risk control strategy; (3) Earnings fluctuation affects the risk control strategy in the same direction, but this effect is more obvious when the fluctuation is slight. When the fluctuation exceeds a certain degree, the fluctuation has little impact on the risk control strategy. The research provides useful guidance for institutional investors to choose investment and risk control strategies.