Risk-sensitive Control and Diierential Games in Innnite Dimensions 1

We study a stochastic risk-sensitive control problem in a bounded subset of an innnite dimensional, separable Hilbert space. We prove that the value functions of stochastic problems are unique viscosity solutions of the Dirichlet boundary value problems for the associated Hamilton-Jacobi-Bellman equations. We show that after a logarithmic transformation the value functions of small noise problems converge uniformly to the unique viscosity solution of the limiting equation. The limit function has an interpretation as the value of a deterministic diierential game. Explicit estimates on the rate of convergence are obtained. 0. Introduction. In this paper we provide a rigorous mathematical treatment of a stochastic risk-sensitive control problem in an innnite dimensional Hilbert space. The problem consists in controling a noisy process so that it does not leave a certain prescribed set. Finite dimensional version of it has been subject of intensive study and a recent paper 8] gives an excellent presentation of the problem together with a