On the First Passage time for Brownian Motion Subordinated by a Lévy Process

Abstract

In this paper we consider the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance-gamma (VG) model, the normal-inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call the first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that the standard first passage time is the almost-sure limit of iterations of the first passage of the second kind. Many different problems arising in financial mathematics are posed as first passage problems, and motivated by this fact, we are led to consider the implications of the approximation scheme for fast numerical methods for computing first passage. We find that the generic form of the iteration can be competitive with other numerical techniques. In the particular case of the VG model, the scheme can be further refined to give very fast algorithms.

Spectral analysis of subordinate Brownian motions on the half-line

“…Noteworthy, the double Laplace transform (in t and x) of the distribution of τ x is known for general Lévy processes since 1957 (Theorem 1 in [4]). Nevertheless, explicit formulae for P x (τ (0,∞) > t) are known only in some special cases; see [24,32,35,51,52] for some recent developments in this area. Also, a formula for the single Laplace transform (in t) of P(τ x > t) for a large class of symmetric Lévy processes was obtained only very recently in [55].…”

First passage times for subordinate Brownian motions

“…A formula for the single Laplace transform for symmetric Lévy processes, under some mild assumptions, was given recently in [29] (see Theorem 1.3 below). In the development of the fluctuation theory for Lévy processes, many new identities involving first passage times were derived (see [6,12,30,33] for a general account on fluctuation theory), including various other characterisations of P(τ x > t), at least in the stable case, see [5,7,10,11,14,15,16,18,19,26,27,36]. First passage times τ x and the supremum functional M t play an important role in many areas of applied probability ( [2,3]), mathematical physics ( [20,25]), and also in potential theory of Lévy processes ( [9,21,22,23,24]).…”

Suprema of Lévy processes

“…1 2 M. KWAŚNICKI, J. MA LECKI AND M. RYZNAR by Darling [11], for a compound Poisson process with Ψ(ξ) = 1 − cos ξ by Baxter and Donsker [3] and for the Poisson process with drift by Pyke [32].The development of the fluctuation theory for Lévy processes resulted in many new identities involving the supremum functional M t ; see, for example, [5,13,31,33]. There are numerous other representations for the distribution of M t , at least in the stable case; see [4,7,11,12,15,16,19,20,27,28,30,36]. The main goal of this article is to give a more explicit formula for P(M t < x) and simple sharp bounds for P(M t < x) in terms of the Lévy-Khintchin exponent Ψ(ξ) for a class of Lévy processes.…”