Singular Control in a Cash Management Model with Ambiguity

TL;DR

This paper develops a singular control model for cash reserve management under ambiguity, revealing how increased ambiguity raises costs and narrows inaction regions, with implications for real-world cash management behavior.

Contribution

It introduces a novel singular control framework incorporating ambiguity via maxmin preferences and establishes a verification theorem for optimal policies.

Findings

Higher ambiguity increases expected costs under worst-case priors.

Ambiguity narrows the inaction region in cash management.

Optimal policies are characterized as control barrier policies.

Abstract

We consider a singular control model of cash reserve management, driven by a diffusion under ambiguity. The manager is assumed to have maxmin preferences over a set of priors characterized by $\kappa$-ignorance. A verification theorem is established to determine the firm's cost function and the optimal cash policy; the latter taking the form of a control barrier policy. In a model driven by arithmetic Brownian motion, we use Dynkin games to show that an increase in ambiguity leads to higher expected costs under the worst-case prior and a narrower inaction region. The latter effect can be used to provide an ambiguity-driven explanation for observed cash management behavior. Our findings can be applied to broader applications of singular control in managing inventories under ambiguity.