Electronic Journal of Probability

Deep factorisation of the stable process

Andreas E. Kyprianou

Full-text: Open access

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 Lévy 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].

Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 23, 28 pp.

Dates
Received: 24 August 2015
Accepted: 17 March 2016
First available in Project Euclid: 5 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1459880111

Digital Object Identifier
doi:10.1214/16-EJP4506

Zentralblatt MATH identifier
1338.60130

Subjects
Primary: 60G52: Stable processes 60G18: Self-similar processes 60G51: Processes with independent increments; Lévy processes

Keywords
stable process Wiener–Hopf factorisation self-similar processes Markov additive processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kyprianou, Andreas E. Deep factorisation of the stable process. Electron. J. Probab. 21 (2016), paper no. 23, 28 pp. doi:10.1214/16-EJP4506. https://projecteuclid.org/euclid.ejp/1459880111


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