On an Optimal Stopping Problem of Time Inhomogeneous Diffusion Processes

For given quasi-continuous functions g, h with g ≤ h and diffusion process M determined by stochastic differential equations or symmetric Dirichlet forms, characterizations of the value functions eg(s, x) = sup σ J (s,x) (σ) and ¯ w(s, x) = infτ sup σ J (s,x) (σ, τ) are well studied so far. In this paper, by using the time dependent Dirichlet forms, we generalize these results to time inhomogeneous diffusion processes. The difficulty of our case arises from the existence of essential semipolar sets. In particular, excessive functions are not necessarily continuous along the sample paths. We get the result by showing such continuity of the value functions.