Large Deviation Approaches for the Numerical Computation of the Hitting Probability for Gaussian Processes

We state large deviations for small time of a pinned n-conditional Gaussian process, i.e. the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants, by letting all the past monitoring instants to depend on the small parameter going to 0. Differently from what already developed in Caramellino and Pacchiarotti (Adv Appl Probab 40:424–453, 2008), this procedure is able to catch the dependence on the past observations. We apply the results to numerical experiments that involve the fractional Brownian motion, for the computation of the hitting probability through Monte Carlo methods.