Paper: arXiv 2505.07761

Authors: Arnon Archankul, Jacco J. J. Thijssen

Abstract

We consider singular control in inventory management under Knightian uncertainty, where decision makers have a smooth ambiguity preference over Gaussian-generated priors. We demonstrate that continuous-time smooth ambiguity is the infinitesimal limit of Kalman-Bucy filtering with recursive robust utility. Additionally, we prove that the cost function can be determined by solving forward-backward stochastic differential equations with quadratic growth. With a sufficient condition and utilising variational inequalities in a viscosity sense, we derive the value function and optimal control policy. By the change-of-coordinate technique, we transform the problem into two-dimensional singular control, offering insights into model learning and aligning with classical singular control free boundary problems. We numerically implement our theory using a Markov chain approximation, where inventory is modeled as cash management following an arithmetic Brownian motion. Our numerical results indicate that the continuation region can be divided into three key areas: (i) the target region; (ii) the region where it is optimal to learn and do nothing; and (iii) the region where control becomes predominant and learning should inactive. We demonstrate that ambiguity drives the decision maker to act earlier, leading to a smaller continuation region. This effect becomes more pronounced at the target region as the decision maker gains confidence from a longer learning period. However, these dynamics do not extend to the third region, where learning is excluded.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical synthesis of singular control, Kalman-Bucy filtering, and quadratic BSDEs under smooth ambiguity preferences. While technically impressive and highly novel theoretically, it relies on stylized numerical approximations (arithmetic Brownian motion via Markov chains) rather than empirical data analysis.

Research Flowchart

  flowchart TD
    A[Research Goal: Inventory Control under Knightian Uncertainty & Smooth Ambiguity] --> B[Theoretical Foundations: Continuous-time Limit of KB-Filtering & Quadratic FBSDEs]
    B --> C[Methodology: Variational Inequalities, Viscosity Solutions & Coordinate Transformations]
    C --> D[Numerical Setup: Cash Management via Arithmetic Brownian Motion & Markov Chain Approximation]
    D --> E[Key Outcomes: 3-Zone Continuation Region & Ambiguity Drives Earlier Control Action]

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