Singular Control in Inventory Management with Smooth Ambiguity

TL;DR

This paper develops a framework for singular inventory control under smooth ambiguity, integrating robust utility, stochastic filtering, and numerical methods to analyze optimal policies and the impact of ambiguity on decision timing.

Contribution

It introduces a novel approach to singular control under smooth ambiguity, linking it with Kalman-Bucy filtering and providing a numerical scheme for implementation.

Findings

Ambiguity causes earlier action and reduces the continuation region.

The continuation region is divided into target, learning, and control zones.

Longer learning periods increase confidence and influence control decisions.

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.