Singular perturbations in risk-sensitive stochastic control

Abstract

Control of nondegenerate diffusions with infinite horizon risk-sensitive criterion is studied when the dynamics exhibits two distinct time scales. If the time scales are separated by a factor $\epsilon>0$, then it is shown that under suitable hypotheses, as $\epsilon\downarrow0$, the optimal cost converges to the optimal risk-sensitive cost for a reduced order controlled diffusion. The dynamics of this diffusion corresponds to the dynamics of the slower variables of the original process, with the dependence on the fast variables averaged out as per the asymptotic behavior of the latter. The arguments use a logarithmic transformation to convert the risk-sensitive control problem into a two-person zero-sum ergodic game, followed by the small parameter asymptotics of the associated Hamilton-Jacobi-Isaacs equation.