Risk-Sensitive Zero-Sum Differential Games

We consider two-player risk-sensitive zero-sum differential games (RSZSDGs). In our problem setup, both the drift term and the diffusion term in the controlled stochastic differential equation are dependent on the state and controls of both players, and the objective functional is of the risk-sensitive type. First, a stochastic maximum principle type necessary condition for an open-loop saddle point of the RSZSDG is established via nonlinear transformations of the adjoint processes of the equivalent risk-neutral stochastic zero-sum differential game. In particular, we obtain two variational inequalities, namely, the pair of saddle-point inequalities of the RSZSDG. Next, we obtain the Hamilton–Jacobi–Isaacs partial differential equation for the RSZSDG, which provides a sufficient condition for a feedback saddle point of the RSZSDG, using a logarithmic transformation of the associated value function. Finally, we study the extended linear-quadratic RSZSDG (LQ-RSZSDG). We show intractability of the extended LQ-RSZSDG with the state and/or controls of both players appearing in the diffusion term. This unexpected intractability could lead to nonlinear open-loop and feedback saddle points even if the problem itself is essentially LQ and the Isaacs condition holds.