Asymptotics of impulse control problem with multiplicative reward

TL;DR

This paper analyzes the long-term behavior of an impulse control problem with multiplicative rewards for Markov processes, providing a solution to the Bellman equation using probabilistic methods and the Krein-Rutman theorem.

Contribution

It introduces a novel approach to solving the Bellman equation for multiplicative reward impulse control problems using probabilistic techniques and spectral theory.

Findings

Constructed a solution to the Bellman equation.

Provided a verification theorem for the control problem.

Utilized approximation methods in bounded domains.

Abstract

We consider a long-run impulse control problem for a generic Markov process with a multiplicative reward functional. We construct a solution to the associated Bellman equation and provide a verification result. The argument is based on the probabilistic properties of the underlying process combined with the Krein-Rutman theorem applied to the specific non-linear operator. Also, it utilises the approximation of the problem in the bounded domain and with the help of the dyadic time-grid.