REFINED SOLUTIONS OF TIME INHOMOGENEOUS OPTIMAL
STOPPING PROBLEM AND ZERO-SUM GAME VIA DIRICHLET FORM

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 process and some other assumptions, the refined solutions without
exceptional starting points are proved to exist, and the value functions of the optimal
stopping problem and zero-sum game, which belong to certain functional spaces,
are characterized as the solutions of some variational inequalities, respectively.

2000 AMS Mathematics Subject Classification: Primary: 60G40; Secondary: 31C25,
49J40, 60J60.

Keywords and phrases: Time inhomogeneous Dirichlet form, optimal stopping,
zero-sum game, variational inequality.