# Long-run risk sensitive dyadic impulse control

## TL;DR

This paper advances long-run risk-sensitive impulse control for unbounded, non-uniformly ergodic Markov processes by integrating geometric drift and local minorisation, establishing solution existence for Bellman equations.

## Contribution

It extends the local span-contraction approach to unbounded processes using weight norms, addressing a gap in the literature on risk-sensitive control.

## Key findings

- Established existence of solutions to Bellman equations for unbounded processes
- Extended geometric drift and minorisation techniques to risk-sensitive impulse control
- Provided examples of processes satisfying the new assumptions

## Abstract

In this paper long-run risk sensitive optimisation problem is studied with dyadic impulse control applied to continuous-time Feller-Markov process. In contrast to the existing literature, focus is put on unbounded and non-uniformly ergodic case by adapting the weight norm approach. In particular, it is shown how to combine geometric drift with local minorisation property in order to extend local span-contraction approach when the process as well as the linked reward/cost functions are unbounded. For any predefined risk-aversion parameter, the existence of solution to suitable Bellman equation is shown and linked to the underlying stochastic control problem. For completeness, examples of uncontrolled processes that satisfy the geometric drift assumption are provided.