Deep factorisation of the stable process

Andreas E Kyprianou

doi:10.1214/16-ejp4506

Abstract

The Lamperti–Kiu transformation for real-valued self-similar Markov processes (rssMp) states that, associated to each rssMp via a space-time transformation, there is a Markov additive process (MAP). In the case that the rssMp is taken to be an $\alpha $-stable process with $\alpha \in (0,2)$, [16] and [24] have computed explicitly the characteristics of the matrix exponent of the semi-group of the embedded MAP, which we henceforth refer to as the Lamperti-stable MAP. Specifically, the matrix exponent of the Lamperti-stable MAP’s transition semi-group can be written in a compact form using only gamma functions. Just as with Levy processes, there exists a factorisation of the (matrix) exponents of MAPs, with each of the two factors uniquely characterising the ascending and descending ladder processes, which themselves are again MAPs. Although the case of MAPs with jumps in only one direction should be relatively straightforward to handle, to the author’s knowledge, not a single example of such a factorisation for two-sided jumping MAPs currently exists in the literature. In this article we provide a completely explicit Wiener–Hopf factorisation for the Lamperti-stable MAP. The main value and novelty of exploring the matrix Wiener–Hopf factorisation of the underlying MAP comes about through style of the computational approach. Understanding the fluctuation theory of the underlying MAP offers new insight into different ways of analysing stable processes. Indeed, we obtain new space-time invariance properties of stable processes, as well as demonstrating examples how new fluctuation identities for stable processes can be developed as a consequence of the reasoning in deriving the matrix Wiener–Hopf factors. The methodology in this paper has already lead to new applications in [27] and [28].

Highlights

A Levy process X is called α-stable if it satisfies the scaling property cXc−αt t≥0 Px =d X |Pcx, c > 0
As Bernstein functions, κ and κcan be seen as the Laplace exponents of subordinators
Observe the process (ξ, J) only at times of increase of new maxima of ξ. This gives a Markov additive processes (MAPs), say (H+(t), J+(t))t≥0, with the property that H is non-decreasing with the same range as the running maximum

Summary

Deep factorisation of the stable process

A Levy process X is called (strictly) α-stable if it satisfies the scaling property cXc−αt t≥0 Px =d X |Pcx , c > 0. [α = 2 → BM, exclude this.] The quantity ρ = P0(Xt ≥ 0) will frequently appear as will ρ = 1 − ρ For a given characteristic exponent of a Levy process, Ψ, there exist unique Bernstein functions, κ and κsuch that, up to a multiplicative constant, Ψ(θ) = κ(iθ)κ(−iθ), θ ∈ R. Hypergeometric Levy processes are another recently discovered family of Levy processes for which the factorisation are known explicitly: For appropriate parameters (β, γ, β, γ). Another factorisation exists, which is more ‘deeply’ embedded in the stable process.

Pathwise description of a MAP

Denote the transition rate matrix of the chain J by

Ascending ladder MAP