Optimal cash management using impulse control

TL;DR

This paper develops a verification theorem for optimal control band policies in impulse control of Levy processes, with applications to cash management involving fixed and proportional costs, analyzing both transient and steady-state behaviors.

Contribution

It provides a new verification theorem for control band policies in Levy process impulse control and explicitly solves for the optimal policy in a specific Levy process case.

Findings

Verification theorem for control band policies

Explicit solution for Levy process with Brownian and Poisson components

Analysis of transient and steady-state behavior

Abstract

We consider the impulse control of Levy processes under the infinite horizon, discounted cost criterion. Our motivating example is the cash management problem in which a controller is charged a fixed plus proportional cost for adding to or withdrawing from his/her reserve, plus an opportunity cost for keeping any cash on hand. Our main result is to provide a verification theorem for the optimality of control band policies in this scenario. We also analyze the transient and steady-state behavior of the controlled process under control band policies and explicitly solve for the optimal policy in the case in which the Levy process to be controlled is the sum of a Brownian motion with drift and a compound Poisson process with exponentially distributed jump sizes.