A Singular Stochastic Control Problem with Interconnected Dynamics

TL;DR

This paper analyzes a complex two-dimensional stochastic control problem with interconnected dynamics, revealing the structure of the value function and free boundaries using advanced mathematical techniques.

Contribution

It introduces a novel analysis of a coupled mean-reverting and controlled process, deriving the structure of the value function and free boundaries with viscosity and free-boundary methods.

Findings

Characterization of the value function via smooth-fit principle

Identification of free boundaries dividing the state space

Derivation of properties of the free boundaries

Abstract

In this paper we study a Markovian two-dimensional bounded-variation stochastic control problem whose state process consists of a diffusive mean-reverting component and of a purely controlled one. The main problem's characteristic lies in the interaction of the two components of the state process: the mean-reversion level of the diffusive component is an affine function of the current value of the purely controlled one. By relying on a combination of techniques from viscosity theory and free-boundary analysis, we provide the structure of the value function and we show that it satisfies a second-order smooth-fit principle. Such a regularity is then exploited in order to determine a system of functional equations solved by the two monotone continuous curves (free boundaries) that split the control problem's state space in three connected regions. Further properties of the free boundaries are also obtained.