Risk-sensitive mean field stochastic differential games

Hamidou Tembine (1) , Quanyan Zhu (2) , Tamer Basar (2)

1 E3S - Supélec Sciences des Systèmes
2 CSL - Coordinated Science Laboratory

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.

Domaines: Gestion des risques [q-fin.RM]; Optimisation et contrôle [math.OC]; Informatique et théorie des jeux [cs.GT]; Théorie [stat.TH]

Hamidou Tembine, Quanyan Zhu, Tamer Basar. Risk-sensitive mean field stochastic differential games. 18th IFAC World Congress, Aug 2011, Milano, Italy. pp.3222-3227, ⟨10.3182/20110828-6-IT-1002.02247⟩. ⟨hal-00643547⟩