Mean-Field Risk-Sensitive Control with HARA Utility

In this paper, we investigate some properties of optimal mean-field Hyperbolic Absolute Risk Aversion (HARA, in short) controllers, and show that the mean-field HARA stochastic optimal control problem is equivalent to a certain mean-field stochastic differential game. Then, we apply those results to a terminal wealth expected utility maximization problem. We consider the investment behavior of investment companies, and help companies in different growth stages (the mature, large investment companies and small, medium and micro companies) find the most optimal investment strategy for themselves. In addition, we study a special linear quadratic case of the mean-field HARA problem and show that finding a robust mean-field HARA controller for a class of linear systems with a quadratic form is equivalent to solving a mean-field linear quadratic problem of the same form, but with greater noise intensity. From the equivalence result we find an interesting relationship between robustness and uncertainty.

1 Introduction

It is known that using the risk-sensitive methodology can help us find robust controllers. Risk-sensitive control is widely used in mathematical finance problems, such as the optimal portfolio choice problem in the financial market [12], the terminal wealth expected utility maximization problem [7], the long-term investment problems [6] and so on. Moreover, in 2007, Lasry and Lions [8] formally introduced the concept of mean-field games by studying stochastic difference games with particles (players) and discussing the asymptotic behavior of these randomly moving particles. In recent years, the idea of mean-field has also penetrated into the economic and financial fields [1].