Stopping Time Problems and the Shape of the Domain of Continuation

Abstract

Let x(t) be an n-dimensional diffusion process defined as a solution of a system of stochastic differential equations

$$dx(t) = \sigma (x(t),t)dw(t) + b(x(t),t)dt$$

((1.1))

with the n-vector b(x,t) and the n × n matrix σ(x,t) continuous in (x,t) ∈ Rⁿx(-∞, ∞) and uniformly Lipschitz continuous in x.