Ergodic Risk Sensitive Control of Diffusions under a General Structural Hypothesis

TL;DR

This paper characterizes optimal stationary Markov controls for ergodic risk-sensitive diffusion processes under a general structural hypothesis, using a variational approach and auxiliary control techniques.

Contribution

It introduces a novel variational formula approach to solve the ergodic risk-sensitive control problem under broad structural conditions.

Findings

Complete characterization of optimal controls under the given hypotheses.

Development of a variational formula for exponential functionals of Brownian motion.

Establishment of a priori estimates for extended diffusions to ensure tightness.

Abstract

We study the infinite-horizon average (ergodic) risk sensitive control problem for diffusion processes under a general structural hypothesis: there is a partition of state space into two subsets, where the controlled diffusion process satisfies a Foster-Lyapunov type drift condition in one subset, under any stationary Markov control, while the near-monotonicity condition is satisfied with the running cost function being inf-compact in its complement. Under these conditions, we completely characterize the optimal stationary Markov controls. To prove this, we consider an inf-compact perturbation to the running cost over the entire space such that the resulting ergodic risk sensitive control problem is well-defined and then use the corresponding existing results. The heart of the analysis lies in exploiting the variational formula of exponential functionals of Brownian motion and applying it to the objective exponential cost function of the controlled diffusion. This representation facilitates us to view the risk sensitive cost for any stationary Markov control as the optimal value of a control problem of an extended diffusion involving a new auxiliary control where the optimal criterion is to maximize the associated long-run average cost criterion that is a difference of the original running cost and an extra term that is quadratic in the auxiliary control. The main difficulty in using this approach lies in the fact that tightness of mean empirical measures of the extended diffusion is not a priori implied by the analogous tightness property of the original diffusion. We overcome this by establishing a priori estimates for the extended diffusion associated with the nearly optimal auxiliary controls.