Refined Solutions of Time Inhomogeneous Optimal Stopping Games via Dirichlet Form

TL;DR

This paper investigates the properties of value functions in time inhomogeneous optimal stopping and Dynkin games using time-dependent Dirichlet forms, establishing refined solutions and characterizations under certain conditions.

Contribution

It introduces refined solutions for these problems without exceptional points, characterized as solutions to variational inequalities, advancing the theoretical understanding of such stochastic games.

Findings

Existence of refined solutions without exceptional starting points.

Value functions are finely and cofinely continuous.

Characterization of value functions via variational inequalities.

Abstract

The properties of value functions of time inhomogeneous optimal stopping problem and zero-sum game (Dynkin game) are studied through time dependent Dirichlet form. Under the absolute continuity condition on the transition function of the underlying diffusion process and some other assumptions, the refined solutions without exceptional starting points are proved to exist, and the value functions of the optimal stopping and zero-sum game, which are finely and cofinely continuous, are characterized as the solutions of some variational inequalities, respectively.