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

Ari Arapostathis and Anup Biswas

Stochastic Processes and their Applications, 2018, vol. 128, issue 5, 1485-1524

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 Rd.

Keywords: Risk-sensitive control; Multiplicative Poisson equation; Controlled diffusions; Nonlinear eigenvalue problems; Hamilton–Jacobi–Bellman equation; Monotonicity of principal eigenvalue (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (4)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:128:y:2018:i:5:p:1485-1524

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DOI: 10.1016/j.spa.2017.08.001

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