Abstract
We consider a class of diffusions controlled through the drift and jump size, and driven by a jump Lévy process and a nondegenerate Wiener process, and we study infinite horizon (ergodic) risk-sensitive control problems for this model. We start with the controlled Dirichlet eigenvalue problem in smooth bounded domains, which also allows us to generalize current results in the literature on exit rate control problems. Then we consider the infinite horizon average risk-sensitive minimization and maximization problems on the whole domain. Under suitable hypotheses, we establish existence and uniqueness of a principal eigenfunction for the Hamilton–Jacobi–Bellman (HJB) operator on the whole space, and fully characterize stationary Markov optimal controls as the measurable selectors of this HJB equation.
Funding Statement
The research of Ari Arapostathis was supported in part by the National Science Foundation through grant DMS-1715210, in part by the Army Research Office through grant W911NF-17-1-001, and in part by the Office of Naval Research through grant N00014-16-1-2956, and was approved for public release under DCN #43-4933-19.
The research of Anup Biswas was supported in part by an INSPIRE faculty fellowship, a SwarnaJayanti fellowship, and DST-SERB grants EMR/2016/004810, MTR/2018/000028.
Acknowledgments
The authors thank the referees for their insightful comments.
Citation
Ari Arapostathis. Anup Biswas. "Risk-sensitive control for a class of diffusions with jumps." Ann. Appl. Probab. 32 (6) 4106 - 4142, December 2022. https://doi.org/10.1214/21-AAP1758
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