Finite Horizon Time Inhomogeneous Singular Control Problem of One-dimensional Diffusion via Dynkin Game

TL;DR

This paper solves a finite horizon time inhomogeneous singular control problem for one-dimensional diffusions using a Dynkin game approach, establishing conditions for classical solutions and characterizing optimal control policies.

Contribution

It introduces a novel method linking Dynkin games to singular control problems, providing explicit conditions for classical solution existence and describing optimal control strategies.

Findings

Value function regularity is established for the Dynkin game.

Optimal control involves reflecting the diffusion between two boundaries.

Impulse control may occur at maturity for general terminal payoffs.

Abstract

The Hamilton-Jacobi-Bellman equation (HJB) associated with the time inhomogeneous singular control problem is a parabolic partial differential equation, and the existence of a classical solution is usually difficult to prove. In this paper, a class of finite horizon stochastic singular control problems of one dimensional diffusion is solved via a time inhomogeneous zero-sum game (Dynkin game). The regularity of the value function of the Dynkin game is investigated, and its integrated form coincides with the value function of the singular control problem. We provide conditions under which a classical solution to the associated HJB equation exists, thus the usual viscosity solution approach is avoided. We also show that the optimal control policy is to reflect the diffusion between two time inhomogeneous boundaries. For a more general terminal payoff function, we showed that the optimal control involves a possible impulse at maturity.