Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Singular Control in Inventory Management with Smooth Ambiguity

Arnon Archankul, Jacco J. J. Thijssen

TL;DR

This paper develops a singular control framework for inventory management under smooth ambiguity, where the decision maker learns about an unknown Gaussian-driven demand process via Kalman–Bucy filtering. The cost functional incorporates a recursive certainty-equivalence utility, leading to a robust, KL-divergence-regularized objective that links to a system of forward–backward SDEs with quadratic growth. A viscosity-solution approach to the associated HJB equation, together with a Johnson–Peskir type coordinate transformation, reduces the problem to a two-dimensional, tractable singular-control problem with identifiable free boundaries. The authors implement a Markov chain approximation to numerically solve the transformed HJB and perform extensive comparative statics, revealing a three-region continuation structure (target, learning-dominant, control-dominant) and showing that higher ambiguity shrinks the continuation region and accelerates intervention, with longer learning periods increasing confidence and narrowing the region near the target.

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.

Singular Control in Inventory Management with Smooth Ambiguity

TL;DR

This paper develops a singular control framework for inventory management under smooth ambiguity, where the decision maker learns about an unknown Gaussian-driven demand process via Kalman–Bucy filtering. The cost functional incorporates a recursive certainty-equivalence utility, leading to a robust, KL-divergence-regularized objective that links to a system of forward–backward SDEs with quadratic growth. A viscosity-solution approach to the associated HJB equation, together with a Johnson–Peskir type coordinate transformation, reduces the problem to a two-dimensional, tractable singular-control problem with identifiable free boundaries. The authors implement a Markov chain approximation to numerically solve the transformed HJB and perform extensive comparative statics, revealing a three-region continuation structure (target, learning-dominant, control-dominant) and showing that higher ambiguity shrinks the continuation region and accelerates intervention, with longer learning periods increasing confidence and narrowing the region near the target.

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.
Paper Structure (19 sections, 10 theorems, 122 equations, 2 figures)

This paper contains 19 sections, 10 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model formulation
  3. Optimisation problem under ambiguity
  4. Heuristic derivation of the optimal control policy
  5. Viscosity Solution of the Value Function
  6. Coordination transform and free boundaries characterisation
  7. Numerical Implementation
  8. Markov chain approximation
  9. Comparative statics of Arithmetic Brownian Motion
  10. General characteristic of the control policy
  11. Comparative Statics of ambiguity attitude
  12. Comparative statics of observation time
  13. Comparative statics of risk
  14. Comparative statics of belief variance
  15. Comparative statics of discounted rate
  16. ...and 4 more sections

Key Result

Lemma 1

The conditional expectation def: distorted cost function satisfies where $h>0$ is chosen to be small enough that $t+h<t+2h<\ldots\leq t+nh=T$, for some $n\in \mathds{N}$. Moreover, if we take $h\to 0$, then

Figures (2)

  • Figure 1: Value functions with fixed parameters $\rho=0.2,a=0.2,b=0.2,s=0.1,\ell=u=2, \check{c}=\hat{c}=1$ and $T=20$ at $\tau=20$. Panel (\ref{['fig:vf_surface']}) displays the surface plots for $\gamma\in\{0,20,40\}$ in an ascending order, while panel (\ref{['fig:vf+cb_contour']}) demonstrates the contour plot and control barriers for $\gamma=20$. Here $\underline \Gamma$, $\overline \Gamma$ and $\Gamma^\ast$ denote the lower barrier, the upper barrier and the minimum points along the process $X_1$, respectively.
  • Figure 2: Control barriers of value functions of terminal time $T=20$.

Theorems & Definitions (16)

  • Lemma 1: Logarithmic Transformation
  • Definition 1
  • Lemma 2: Exponential Martingale
  • Lemma 3: Comparison Theorem
  • Theorem 1
  • Lemma 4: Dynamic Programming Principle
  • Definition 2: Viscosity Solution
  • Remark 1
  • Definition 3
  • Theorem 2
  • ...and 6 more