On the first-passage times of certain Gaussian processes, and related asymptotics

The first-passage time sSðxÞ of a one-dimensional continuous stochastic process XðtÞ, starting from x Sð0Þ, through a smooth boundary S(t) is investigated; in particular, diffusions and some kinds of Gaussian processes, such as Gauss-Markov and their fractional integrals, are considered. The tail behavior of Pðmaxs2½0, tXðsÞ > RÞ and related asymptotics for sSðxÞ are obtained, and some examples are reported