Suprema of Lévy processes

Abstract

In this paper we study the supremum functional Mt = sup 0≤s≤t Xs, where Xt, t ≥ 0, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of Mt. In the symmetric case we find an integral representation of the Laplace transform of the distribution of Mt if the Lévy-Khintchin exponent of the process increases on (0, ∞). . This reprint differs from the original in pagination and typographic detail. 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. Most estimates of the cumulative distribution function of M t are proved for very general Lévy processes, without symmetry assumptions.Let τ x denote the first passage time through a barrier at the level x for the process X t ,with the infimum understood to be infinity when the set is empty. We always assume that X 0 = 0. Since P(M t < x) = P(τ x > t), the problems of finding the cumulative distribution functions of M t and τ x are the same. The supremum functional and first passage time statistics are important in various areas of applied probability [1,2], as well as in mathematical physics [21,26]. The recent progress in the potential theory of Lévy processes is, in part, due to the application of fluctuation theory; see [9,10,18,[22][23][24][25]. The paper is organized as follows. Section 2 contains some preliminary material related to Bernstein functions, Stieltjes functions and estimates for the Laplace transform. In Section 3 (Theorem 3.1 and Corollary 3.2) we prove, under mild assumptions, the estimate P(M t < x) ≈ min(1, κ(1/t, 0)V (x)), t, x > 0,

Green function estimates for subordinate Brownian motions: Stable and beyond

“…See, for example, [KMR,KSV12a,KSV12b,KSV12c,RSV06] and the references therein. Recently there has been huge interest in studying the potential theory of such processes.…”

“…See, for example, [KMR,KSV12a,KSV12b,KSV12c,RSV06] and the references therein. In this paper, when the Laplace exponent φ of the corresponding subordinator satisfies some mild conditions, we first prove the scale invariant boundary Harnack inequality for X on arbitrary open sets.…”

Record statistics for multiple random walks

“…(24) is known in the literature as the celebrated Pollaczek-Spitzer formula [36,37] and has been used in a number of works to derive exact results on the maximum of a random jump process [23,[38][39][40]. Interestingly, this formula has also been useful to compute the asymptotic behavior of the flux of particles to a spherical trap in 3D [24,41,42].…”

The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

“…Our first result gives the expression for the Laplace transform of q t (dx) (for fixed t > 0) in the case of symmetric Lévy processes with increasing Lévy-Khintchin exponent. This is an analogue of Theorem 4.1 in [15], where the corresponding formula for X t was derived. Note that even though the formulae for the Laplace transforms of q t (dx) and P(X t ∈ dx) seem to be similar, passing from one to the other by using (2.3) and (3.1) is not straightforward.…”

“…Proof. The proof is based on the same idea as in the proof of Theorem 4.1 in [15] with a slight modification of the arguments. For the completeness of the exposure and the convenience of the reader we present it below.…”