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

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 ${\mathbb{R}}^d$.

37 pages

References in corpus (4)

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• On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control
• Downside risk minimization via a large deviations approach
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Cited by in corpus (8)

• Strict monotonicity of principal eigenvalues of elliptic operators in ${\mathbb{R}}^d$ and risk-sensitive control
• Ergodic Risk-Sensitive Control of Markov Processes on Countable State Space Revisited
• A variational characterization of the risk-sensitive average reward for controlled diffusions on ${\mathbb{R}}^d$
• A variational formula for risk-sensitive control of diffusions in ${\mathbb{R}}^d$
• Risk sensitive optimal stopping
• On the policy improvement algorithm for ergodic risk-sensitive control
• Existence of bounded solutions to multiplicative Poisson equations under mixing property
• Asymptotics of impulse control problem with multiplicative reward