Abstract
This article is about right inverses of Levy processes as first introduced by Evans in the symmetric case and later studied systematically by the present authors and their co-authors. Here we add to the existing fluctuation theory an explicit description of the excursion measure away from the (minimal) right inverse. This description unifies known formulas in the case of a positive Gaussian coefficient and in the bounded variation case. While these known formulas relate to excursions away from a point starting negative continuously, and excursions started by a jump, the present description is in terms of excursions away from the supremum continued up to a return time. In the unbounded variation case with zero Gaussian coefficient previously excluded, excursions start negative continuously, but the excursion measures away from the right inverse and away from a point are mutually singular. We also provide a new construction and a new formula for the Laplace exponent of the minimal right inverse.
Highlights
Evans [5] defined a right inverse of a Lévy process X = (X t, t ≥ 0) to be any increasing process K = (Kx, x ≥ 0) such that XKx = x for all x ≥ 0
The existence of partial right inverses is a local path property that has been completely characterised [4, 5, 8] in terms of the Lévy-Khintchine triplet (a, σ2, Π) of the Lévy process X, i.e
((X t∧ζ(X ), t ≥ 0) ∈ · |X0 = y) as canonical measure on E ⊂ D of the distribution of X starting from y and frozen when hitting zero
Summary
A partial right inverse [8] is any increasing process K = (Kx , 0 ≤ x < ξK ) such that XKx = x for all 0 ≤ x < ξK for some (random) ξK > 0. The existence of partial right inverses is a local path property that has been completely characterised [4, 5, 8] in terms of the Lévy-Khintchine triplet (a, σ2, Π) of the Lévy process X , i.e. a ∈ , σ2 ≥ 0 and Π measure on with Π({0}) = 0 and (1 ∧ y2)Π(d y) < ∞ such that eiλX t = e−tψ(λ), where ψ(λ) = −iaλ + 1 σ2λ2 + 2.
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