Finite Time-Horizon Risk Sensitive Control and the Robust Limit under a Quadratic Growth Assumption

The finite time--horizon risk sensitive limit problem for continuous, nonlinear systems is considered. Previous results are extended to cover more typical examples. In particular, the cost may grow quadratically, and the diffusion coefficient may depend on the state. It is shown that the risk sensitive value function is the solution of the corresponding dynamic programming equation. It is also shown that this value converges to the value of the Robust control problem as the cost becomes infintely risk averse with corresponding scaling of the diffusion coefficient. Key Words: risk-sensitive control, robust, H1 , viscosity solutions, nonlinear HJB equations, nonlinear Isaacs equations AMS subject classifications: 35B37, 49L25, 90D25, 93B36, 93C10, 93E05, 93E20 1 Introduction The nonlinear, finite time--horizon risk sensitive limit problem is considered. It is, by now, well known that the value functions of risk--sensitive stochastic control problems tend to converge to the value functio..