Parisian ruin over a finite-time horizon

For a risk process R (u) (t) = u + ct - X(t), t a parts per thousand yen 0, where u a parts per thousand yen 0 is the initial capital, c > 0 is the premium rate and X(t), t a parts per thousand yen 0 is an aggregate claim process, we investigate the probability of the Parisian ruin P-S(u, T-u) = P{inf(t is an element of[0,S])sup(s is an element of[t,t+Tu]) R-u(s) < 0}, S,T-u > 0. For X being a general Gaussian process we derive approximations of PS(u, T (u) ) as u -> a. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval.