Diffusive limit approximation of pure-jump optimal stochastic control problems

TL;DR

This paper studies the diffusive limit of pure-jump stochastic control problems, showing convergence rates and correction methods, which improve numerical approximations in high-intensity jump scenarios, exemplified by an advertising auction case.

Contribution

It introduces a novel approach to approximate pure-jump control problems via diffusive limits, including convergence analysis and correction term construction.

Findings

Convergence speed depends on the Hessian's Hölder constant.

Correction terms enhance approximation accuracy.

Method is effective in high jump intensity scenarios.

Abstract

We consider the diffusive limit of a typical pure-jump Markovian control problem as the intensity of the driving Poisson process tends to infinity. We show that the convergence speed is provided by the H\"older constant of the Hessian of the limit problem, and explain how correction terms can be constructed. This provides an alternative efficient method for the numerical approximation of the optimal control of a pure-jump problem in situations with very high intensity of jump. We illustrate this approach in the context of a display advertising auction problem.