Long-run risk sensitive impulse control

TL;DR

This paper develops a probabilistic framework for solving long-run risk-sensitive impulse control problems in continuous-time Markov processes, establishing links with Bellman equations and approximation methods.

Contribution

It introduces a novel probabilistic approach to solve continuous-time risk-sensitive impulse control and demonstrates how discretized strategies approximate optimal solutions.

Findings

Solution to continuous-time Bellman equation established

Discretized strategies effectively approximate continuous control

Framework applicable to various embedded processes

Abstract

In this paper we consider long-run risk sensitive average cost impulse control applied to a continuous-time Feller-Markov process. Using the probabilistic approach, we show how to get a solution to a suitable continuous-time Bellman equation and link it with the impulse control problem. The optimal strategy for the underlying problem is constructed as a limit of dyadic impulse strategies by exploiting regularity properties of the linked risk sensitive optimal stopping value functions. In particular, this shows that the discretized setting could be used to approximate near optimal strategies for the underlying continuous time control problem, which facilitates the usage of the standard approximation tools. For completeness, we present examples of processes that could be embedded into our framework.