A Risk-Sensitive Maximum Principle for Non-Zero Sum Differential Games of BSDEs

Abstract

We consider a risk-sensitive non-zero sum game problem of the stochastic system derived by backward stochastic differential equations (BSDEs). The domain of the control is assumed to be convex. Necessary conditions in the form of Pontryagin's maximum principle for open-loop equilibrium point are obtained for the risk-sensitive backward non-zero sum game problem. An interesting feature is that the necessary conditions for the equilibrium point do not depend on the risk-sensitive parameter. We also obtain a verification theorem, which provides a sufficient condition for the equilibrium point. A backward stochastic linear-quadratic risk-sensitive game problem is given to illustrate the theoretical results.