Hybrid Risk-Sensitive Mean-Field Stochastic Differential Games with Application to Molecular Biology

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

1 University of Illinois - Department of Electrical and Computer Engineering [Urbana]
2 E3S - Supélec Sciences des Systèmes

We consider a class of mean-field nonlinear stochastic differential games (resulting from stochastic differential games in a large population regime) with risk-sensitive cost functions and two types of uncertainties: continuous-time disturbances (of Brownian motion type) and event-driven random switching. Under some regularity conditions, we first study the best response of the players to the mean field, and then characterize the (strongly time-consistent Nash) equilibrium solution in terms of backward-forward macroscopic McKean-Vlasov (MV) equations, Fokker-Planck-Kolmogorov (FPK) equations, and Hamilton-Jacobi-Bellman (HJB) equations. We then specialize the solution to linear-quadratic mean-field stochastic differential games, and study in this framework the optimal transport of the GlpF transmembrane channel of Escherichia coli, where glycerol molecules (as players in the game) choose forces to achieve optimal transport through the membrane. Simulation studies show that GlpF improves the glycerol conduction more in a higher periplasmic glycerol concentration, which is consistent with observations made in the biophysics literature.

Mots clés: Game theory; Biomolecular systems; Stochastic optimal control

Domaines: Autre [q-bio.OT]; Applications [stat.AP]; Optimisation et contrôle [math.OC]

Quanyan Zhu, Hamidou Tembine, Tamer Basar. Hybrid Risk-Sensitive Mean-Field Stochastic Differential Games with Application to Molecular Biology. 2011 50th IEEE Conference on Decision and Control and European Control Conference , Dec 2011, Orlando, United States. 6 p., ⟨10.1109/CDC.2011.6161035⟩. ⟨hal-00643490⟩