Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions

TL;DR

This paper addresses the existence and optimality of solutions to the risk-sensitive control problem for diffusions over an infinite horizon, relaxing stability assumptions and providing new theoretical insights into the associated Hamilton-Jacobi-Bellman equation.

Contribution

It establishes existence, optimality, and uniqueness results for the risk-sensitive HJB equation under minimal stability assumptions and introduces new findings on the multiplicative Poisson equation.

Findings

Existence of solutions to the risk-sensitive HJB equation.

Optimality of stationary Markov controls under near-monotonic costs.

Conditions for uniqueness and verification of solutions.

Abstract

We consider the infinite horizon risk-sensitive problem for nondegenerate diffusions with a compact action space, and controlled through the drift. We only impose a structural assumption on the running cost function, namely near-monotonicity, and show that there always exists a solution to the risk-sensitive Hamilton-Jacobi-Bellman (HJB) equation, and that any minimizer in the Hamiltonian is optimal in the class of stationary Markov controls. Under the additional hypothesis that the coefficients of the diffusion are bounded, and satisfy a condition that limits (even though it still allows) transient behavior, we show that any minimizer in the Hamiltonian is optimal in the class of all admissible controls. In addition, we present a sufficient condition, under which the solution of the HJB is unique (up to a multiplicative constant), and establish the usual verification result. We also present some new results concerning the multiplicative Poisson equation for elliptic operators in $\mathbb{R}^d$.