Ergodic control of a class of jump diffusions with finite Lévy measures and rough kernels

Ari Arapostathis, Luis Caffarelli, Guodong Pang, Yi Zheng

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

We study the ergodic control problem for a class of jump diffusions in Rd which are controlled through the drift with bounded controls. The Lévy measure is finite, but has no particular structure; it can be anisotropic and singular. Moreover, there is no blanket ergodicity assumption for the controlled process. Unstable behavior is “discouraged” by the running cost which satisfies a mild coercive hypothesis (i.e., is near-monotone). We first study the problem in its weak formulation as an optimization problem on the space of infinitesimal ergodic occupation measures and derive the Hamilton–Jacobi–Bellman equation under minimal assumptions on the parameters, including verification of optimality results, using only analytical arguments. We also examine the regularity of invariant measures. Then, we address the jump diffusion model and obtain a complete characterization of optimality.

Original languageEnglish (US)
Pages (from-to)1516-1540
Number of pages25
JournalSIAM Journal on Control and Optimization
Volume57
Issue number2
DOIs
StatePublished - Jan 1 2019

All Science Journal Classification (ASJC) codes

  • Control and Optimization
  • Applied Mathematics

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