First passage times for subordinate Brownian motions

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