Martingale optimization problems arising in game theory, applications to financial models

Résumé

We study the asymptotic behavior of a general class of zero-sum repeated games with incomplete information on one side. A typical example is a financial exchange game where an informed agent is facing a sequence of uninformed risk-neutral agents. The principal assumption we need is that the mechanism of the game is such that the informed agent benefits only of the asymmetry of information. Our main result is a characterization of the limiting distributions of any sequence of equilibrium price processes when the number of repetition becomes large. More formally, the value of the finitely repeated games is expressed as an optimization problem over discrete-time martingale distributions representing price processes, and our asymptotic analysis leads us to a continuous-time optimization problem over martingale distributions. The optimal martingales in this limit problem are maximizing an integral cost depending on the instantaneous covariance of the process. A naive interpretation is that the informed agent tries to maximize the "random fluctuations" of the prices. We characterize the solutions of this limit problem using convex duality. The dual problem is a stochastic control problem, whose value is given by the solution of a second-order HJB equation. We prove a "verification" result that allows to solve explicitely the primal problem in some particular cases.