Infinite Horizon Impulse Control of Stochastic Functional Differential Equations

TL;DR

This paper develops a probabilistic framework for optimal impulse control of stochastic functional differential equations driven by Lévy processes, applicable to both infinite and random horizons, using Snell envelopes.

Contribution

It introduces a novel probabilistic approach for establishing the existence of optimal controls for SFDEs with impulse controls over infinite horizons.

Findings

Existence of optimal impulse controls for SFDEs under Lévy noise.

Applicability to both infinite and random horizon settings.

Use of Snell envelopes for the control problem.

Abstract

We consider impulse control of stochastic functional differential equations (SFDEs) driven by L\'evy processes under an additional $L^p$-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.