A variational formula for risk-sensitive control of diffusions in ${\mathbb{R}}^d$

Abstract

We address the variational problem for the generalized principal eigenvalue on ${\mathbb{R}}^d$ of linear and semilinear elliptic operators associated with nondegenerate diffusions controlled through the drift. We establish the Collatz-Wielandt formula for potentials that vanish at infinity under minimal hypotheses, and also for general potentials under blanket geometric ergodicity assumptions. We also present associated results having the flavor of a refined maximum principle.

arXiv comment19 pages

References in corpus (4)

• Strict monotonicity of principal eigenvalues of elliptic operators in ${\mathbb{R}}^d$ and risk-sensitive control
• Infinite horizon risk-sensitive control of diffusions without any blanket stability assumptions
• Risk-Sensitive Control and an Abstract Collatz-Wielandt Formula
• The Dirichlet problem for the Bellman equation at resonance

Cited by in corpus (4)

• A variational characterization of the risk-sensitive average reward for controlled diffusions on ${\mathbb{R}}^d$
• Long-run risk sensitive impulse control
• On the policy improvement algorithm for ergodic risk-sensitive control
• Generalized principal eigenvalues of convex nonlinear elliptic operators in ${\mathbb{R}}^N$