First passage times for subordinate Brownian motions

TL;DR

This paper derives integral formulas and asymptotic behaviors for the tail distribution of first passage times of subordinate Brownian motions with certain Levy measures, providing insights into their probabilistic properties.

Contribution

It introduces new integral formulas for the tail distribution of first passage times and analyzes their asymptotic behavior under mild conditions.

Findings

Integral formulas for P(τ_x > t) and its derivatives

Asymptotic analysis as t→∞ or x→0

Conditions on Levy measure density

Abstract

Let X_t be a subordinate Brownian motion, and suppose that the Levy measure of the underlying subordinator has completely monotone density. Under very mild conditions, we find integral formulae for the tail distribution P(\tau_x > t) of first passage times \tau_x through a barrier at x > 0, and its derivatives in t. As a corollary, we examine the asymptotic behaviour of P(\tau_x > t) and its t-derivatives, either as t goes to infinity or x goes to 0.