Wendell F. Fleming
Professor Emeritus, Division of Applied Mathematics, Brown University
Differential Games and Risk Sensitive Stochastic Control
The first part of this lecture gives a concise historical overview of two-player, zero sum differential games, beginning with the pioneering work of Isaacs. The Crandall-Lions theory of viscosity solutions for nonlinear partial differential equations has a crucial role in the treatment of value functions for differential games.
Differential games provide a convenient link between deterministic and stochastic approaches to control systems. A second part of the lecture concerns ideas of risk sensitivity, large deviations for small random perturbations of dynamical systems and stochastic control. As a measure of risk aversion tends to infinity, a differential game is obtained as the limit of corresponding risk sensitive stochastic control problems. This game can be interpreted as a kind of "max-plus" stochastic control problem, in which the usual operations of addition and multiplication are replaced by their max-plus counterparts.
An example from mathematical finance is given to illustrate these methods.