A Correction to "Dynkin Games Via Dirichlet Forms and Singular Control of One-Dimensional Diffusion"

TL;DR

This paper addresses issues in proving the existence of smooth value functions and optimal policies in a one-dimensional stochastic singular control problem, highlighting necessary conditions or process modifications for validity.

Contribution

It corrects and clarifies the conditions needed for the main theorem in the original work on Dynkin games and singular control of diffusions.

Findings

Identifies missing conditions for the main theorem's validity

Proposes alternative diffusion processes for the theorem to hold

Clarifies the assumptions needed for existence of smooth value functions

Abstract

In the paper "Dynkin Games Via Dirichlet Forms and Singular Control of One-Dimensional Diffusion", the authors tried to show the existences of a smooth value function and an optimal policy to a one-dimensional stochastic singular control problem, where the underlying process is a generalized diffusion process given by dX_t=\mu(X_t)dt+\sigma(X_t)dw_t, in which w_t is a Wiener process. It is found that either a condition on \mu(x) and \sigma(x) should be added, or a different diffusion process should be considered and as a result, the main theorem of this paper should be amended.