A new integral equation for Brownian stopping problems with finite time horizon

For classical finite time horizon stopping problems driven by a Brownian motion V(t,x)=supt≤τ≤0E(t,x)[g(τ,Wτ)],we derive new class of Fredholm type integral equations for the set. large discounted problems, we show analytical arguments that equation uniquely characterizes boundary problem. Regardless uniqueness, use representation to rigorously find limit behavior close terminal time. Interestingly, it turns out leading-order coefficient is universal wide classes problems. We also discuss how can be used numerical purposes.