Singular Control in a Cash Management Model with Ambiguity

Paper: arXiv 2309.12014

Authors: Arnon Archankul, Giorgio Ferrari, Tobias Hellmann, Jacco J. J. Thijssen

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 $κ$-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.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework utilizing singular control, dynamic programming, and Dynkin games to model cash management under Knightian uncertainty. However, it is purely theoretical with no empirical backtesting or real-world data validation included in the summary.

Research Flowchart

  flowchart TD
    A[Research Goal: Cash Reserve Management under Ambiguity] --> B[Inputs: Arithmetic Brownian Motion & Maxmin Preferences]
    B --> C[Methodology: Singular Control & Dynkin Games]
    C --> D[Computational Process: Prove Verification Theorem & Establish Worst-Case Prior]
    D --> E[Outcome 1: Optimal Barrier Policy Formulated]
    D --> F[Outcome 2: Ambiguity Increases Costs & Narrows Inaction Region]