Diffusive limit approximation of pure jump optimal ergodic control problems

TL;DR

This paper investigates the diffusive limit of pure jump ergodic control problems, providing error bounds, correction methods, and numerical strategies to improve computational efficiency in reinforcement learning contexts.

Contribution

It extends existing results to ergodic problems, introduces a first order error correction, and quantifies numerical errors from finite difference schemes.

Findings

Error governed by Hessian Hölder continuity

Constructed a first order error correction term

Quantified numerical error from finite difference schemes

Abstract

Motivated by the design of fast reinforcement learning algorithms, we study the diffusive limit of a class of pure jump ergodic stochastic control problems. We show that, whenever the intensity of jumps is large enough, the approximation error is governed by the H{\"o}lder continuity of the Hessian matrix of the solution to the limit ergodic partial differential equation. This extends to this context the results of [1] obtained for finite horizon problems. We also explain how to construct a first order error correction term under appropriate smoothness assumptions. Finally, we quantify the error induced by the use of the Markov control policy constructed from the numerical finite difference scheme associated to the limit diffusive problem, this seems to be new in the literature and of its own interest. This approach permits to reduce very significantly the numerical resolution cost.