Discrete time stochastic impulse control with delay

TL;DR

This paper develops a probabilistic framework for solving infinite-horizon discrete-time stochastic impulse control problems with execution delays, introducing optimal and near-optimal strategies for risk-neutral and risk-sensitive utilities.

Contribution

It introduces a novel probabilistic approach using Snell envelopes to handle execution delays in discrete-time stochastic impulse control problems.

Findings

Existence of optimal strategies under delay conditions

Construction of bounded epsilon-optimal strategies

Applicability to risk-sensitive utility functions

Abstract

We study a class of infinite-horizon impulse control problems with execution delay in discrete time. Using probabilistic methods, particularly the notion of the Snell envelope of processes, we construct an optimal strategy among all admissible strategies for both risk-neutral and risk-sensitive utility functions. Furthermore, we establish the existence of bounded $\epsilon$-optimal strategies. This framework provides a robust approach to handling execution delays in discrete-time stochastic systems.