Large deviation estimates of the crossing probability for pinned Gaussian processes

Abstract

The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay innfixed points atnfixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions andm-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.