A refined large deviation principle for Brownian motion and its application to boundary crossing

Let W denote standard Brownian motion. We consider large deviations for [var epsilon]1/2W as [var epsilon] tends to zero. Let q be a nondecreasing function on [0, 1] which belongs to the upper class of Brownian motion at the origin. We show that in the usual large deviation principle (see Varadhan, 1984) the uniform topology can be replaced by the topology induced by the q-metric dq(x, y):=sup0 < t <= 1 { x(t) - y(t)/q(t)}. This modification is motivated by an application to boundary crossing probabilities.