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

Abstract

The first-passage time ${\tau }_S(x)$ of a one-dimensional continuous stochastic process $X(t),$ starting from $x\le S(0),$ 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 $P({\max }_{s\in [0,t]}X(s)>R)$ and related asymptotics for ${\tau }_S(x)$ are obtained, and some examples are reported.

KEYWORDS:

• First-passage time
• diffusion
• Gaussian process
• Gauss-Markov process

MATHEMATICS SUBJECT CLASSIFICATION:

• 60J60
• 60H05
• 60H10

Acknowledgments

The author acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.