Risk-sensitive mean field stochastic differential games
Abstract
In this paper, we study a class of risk-sensitive mean-field stochastic di fferential games. Under regularity assumptions, we use results from standard risk-sensitive di fferential game theory to show that the mean- field value of the exponentiated cost functional coincides with the value function of a Hamilton-Jacobi-Bellman-Fleming (HJBF) equation with an additional quadratic term. We provide an explicit solution of the mean- field best response when the instantaneous cost functions are log-quadratic and the state dynamics are affine in the control. An equivalent mean- field risk-neutral problem is formulated and the corresponding mean-fi eld equilibria are characterized in terms of backward-forward macroscopic McKean-Vlasov equations, Fokker- Planck-Kolmogorov equations and HJBF equations.