Upper and lower functions for diffusion processes (Q1113198)

This paper states and proves an integral test for determining whether as \(t\to \infty\) the probability is 0 or is 1 that a given nonnegative nondecreasing function h(t) is less than the solution X(t) of the stochastic differential equation \[ dX(t)=a(X(t),t)dt+dW(t), \] where W(t) is a standard Wiener process and a(x,t) is a nonnegative measurable function which satisfies certain growth and Lipschitz conditions with respect to x.