Stochastic Impulse Control of Non-Markovian Processes

TL;DR

This paper develops a framework for stochastic impulse control of non-Markovian processes, establishing the existence of optimal controls using advanced probabilistic techniques like reflected BSDEs and the Snell envelope.

Contribution

It introduces methods to prove the existence of optimal impulse controls for non-Markovian processes, extending beyond traditional Markovian approaches.

Findings

Existence of optimal impulse control for non-Markovian processes.

Existence of combined stochastic and impulse control for diffusions with random coefficients.

Novel use of reflected BSDEs and the Snell envelope in this context.

Abstract

We consider a class of stochastic impulse control problems of general stochastic processes i.e. not necessarily Markovian. Under fairly general conditions we establish existence of an optimal impulse control. We also prove existence of combined optimal stochastic and impulse control of a fairly general class of diffusions with random coefficients. Unlike, in the Markovian framework, we cannot apply quasi-variational inequalities techniques. We rather derive the main results using techniques involving reflected BSDEs and the Snell envelope.