Hybrid risk-sensitive mean-field stochastic differential games with application to molecular biology

Abstract

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: continuoustime 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 meanfield 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.

Decomposition and Mean-Field Approach to Mixed Integer Optimal Compensation Problems

“…A similar reasoning applies tox 1 (k), used in the evolution of d + 2 (k) as in the second equation of (31). We now observe a negative contribution of the term −κx 1 (k) on d + 2 (k) and therefore take forx 1 (k) a possible lower bound of x 1 (k) as shown in the first equation of (32).…”

“…From a mathematical point of view, the mean-field approach leads to the study of a system of partial differential equations (PDEs), where the classical Hamilton-Jacobi-Bellman equation is coupled with a Fokker-Planck equation for the density of the players, in a forward-backward fashion. The decomposition method proposed here requires that each agent i computes in advance the time evolution of the local average (see, e.g., the Fokker-Planck-Kolmogorov equation in [23,[28][29][30][31][32][33]). However, since this is practically impossible, we use here the predictive control method to approximate the computation of the solution.…”

“…Regarding variablex 2 (k), this is used in the evolution of d + 1 (k) as in the first equation of (31). Because of the positive contribution of the term κx 2 (k) on d + 1 (k), a conservative approach would suggest to take forx 2 (k) a possible upper bound of x 2 (k) and this is exactly the spirit behind the evolution ofx 2 (k) as expressed in the second equation of (32). Here,x 1 is an average value for x 1 .…”

Mixed integer optimal compensation: Decompositions and mean-field approximations

“…From a mathematical point of view, the Mean Field approach leads to a study of a system of PDEs, where the classical Hamilton-Jacobi-Bellman equation is coupled with a Fokker-Planck equation for the density of the players, in an interesting forward-backward way. The decomposition method proposed here requires that each agent i computes in advance the time evolution of the local average (see, e.g., the Fokker-Planck-Kolmogorov equation in [2], [12], [16], [17]). However, since this is practically impossible, we use here the predictive control method to approximate the computation of the solution.…”

Mean-Field Games and Dynamic Demand Management in Power Grids

“…The second PDE is the Fokker-Planck equation which describes the density of the players and is solved forwards in time with boundary conditions on the initial population distribution (see, e.g., the Fokker-Planck-Kolmogorov equation in [2,25,32,36] and in the lecture notes [13] and M.S. thesis [27]).…”