## Electronic Journal of Probability

### Deep factorisation of the stable process

Andreas E. Kyprianou

#### 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 (2016), paper no. 23, 28 pp.

Dates
Accepted: 17 March 2016
First available in Project Euclid: 5 April 2016

https://projecteuclid.org/euclid.ejp/1459880111

Digital Object Identifier
doi:10.1214/16-EJP4506

Mathematical Reviews number (MathSciNet)
MR3485365

Zentralblatt MATH identifier
1338.60130

#### 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

#### References

• [1] http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric2F1/03/03/04/
• [2] G. Alsmeyer : On the Markov renewal theorem. Stoch. Proc. Appl. 50, 37-56, 1994.
• [3] G. Alsmeyer : Quasistochastic matrices and Markov renewal theory. J. Appl. Probab. 51A, 359-376, 2014.
• [4] E. Arjas and T. P. Speed : Symmetric Wiener-Hopf factorisations in Markov Additive Processes.pdf Z.W., 26, 105-118, 1973.
• [5] S. Asmussen: Applied Probability Queues. 2nd Edition. Springer-Verlag, New York, second edition, 2003.
• [6] S. Asmussen and H. Albrecher : Ruin probabilities, volume 14 of Advanced Series on Statistical Science & Applied Probability. World Scientific Publishing Co. Pte. Ltd., Singapore, 2010.
• [7] J. Bertoin : Lévy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996.
• [8] J. Bertoin and R. A. Doney : Cramér’s estimate for Lévy processes. Statist. Probab. Lett. 21 363-365, 1994.
• [9] N. H. Bingham : Fluctuation Theory in Continuous Time. Adv. Appl. Prob., 7, 705-766, 1975.
• [10] R. M. Blumenthal, R. K. Getoor and D. B. Ray : On the distribution of first hits for the symmetric stable process. Trans. Amer. Math. Soc. 99, 540-554, 1961.
• [11] T. Bogdan and T. Żak : On Kelvin Transformation. J. Theor. Probab., 19, 89-120, 2006.
• [12] R. M. Blumenthal and R. K. Getoor : Markov processes and potential theory. Pure and Applied Mathematics, Vol. 29. Academic Press, New York, 1968.
• [13] L. Chaumont, A. E. Kyprianou, J. C. Pardo and V. Rivero : Fluctuation theory and exit systems for positive self-similar Markov processes. Ann. Probab. 40, 245-279, 2012.
• [14] E. Çinlar : Markov additive processes II Z.W., 24, 95-121, 1972.
• [15] E. Çinlar : Lévy systems of Markov additive processes Z.W., 31, 175-185, 1975.
• [16] L. Chaumont, H. Pantí, and V. Rivero : The Lamperti representation of real-valued self-similar Markov processes. Bernoulli, 19, 2494-2523, 2013.
• [17] O. Chybiryakov : The Lamperti correspondence extended to Lévy processes and semi-stable Markov processes in locally compact groups. Stochastic Process. Appl., 116, 857-872, 2006.
• [18] S. Dereich, L. Döring and A. E. Kyprianou : Self-similar Markov processes started from the origin. Preprint.
• [19] J. Ivanovs : One-sided Markov additive processes and related exit problems. PhD thesis, Universiteit van Amsterdam, 2011.
• [20] H. Kesten : Renewal Theory for Functionals of a Markov Chain with General State Space. Ann. Probab. 3, 355-386, 1974.
• [21] H. Kaspi : On the Symmetric Wiener-Hopf Factorization for Markov Additive Processes. Z. Wahrsch. verw. Gebiete, 59, 179-196, (1982).
• [22] P. Klusik and Z. Palmowski : A Note on Wiener–Hopf Factorization for Markov Additive Processes. J. Theor. Probab. 27, 202-219, 2014.
• [23] A. Kuznetsov and J. C. Pardo : Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes Acta Appl. Math., 123, 113-139, 2013.
• [24] A. Kuznetsov, A. E. Kyprianou, J. C. Pardo and A. R. Watson : The hitting time of zero for a stable process. Electron. J. Probab. 19, 1-26, 2014.
• [25] A. E. Kyprianou, J. C Pardo and A. R. Watson : Hitting distributions of $\alpha$-stable processes via path censoring and self-similarity. Ann. Probab. 42, 398-430, 2014.
• [26] A. E. Kyprianou : Introductory lectures on fluctuations of Lévy processes with applications. Universitext. Springer-Verlag, Berlin, 2006.
• [27] A. E. Kyprianou, V. Rivero and B. Şengül : Deep factorisation of the stable process II; potentials and applications. Preprint. arXiv:1511.06356 [math.PR].
• [28] A. E. Kyprianou, V. Rivero and W. Satitkanitkul : Conditioned real self-similar Markov processes . Preprint, arXiv:1510.01781 [math.PR].
• [29] S. P. Lalley : Conditional Markov renewal theory I. Finite and denumerable state space. Ann. Probab., 12, 1113-1148, 1984.
• [30] M. Reisz : Intégrales de Riemann-Liouville et potentiels. Acta. Sci. Math. Szeged. 9, 1-42, 1938.
• [31] M. Reisz : Rectification au travail “Intégrales de Riemann-Liouville et potentiels”. Acta Sci. Math. Szeged. 9, 116-118, 1938.
• [32] B. A. Rogozin : Distribution of the position of hit for stable and asymptotically stable random walks on an interval. Teor. Verojatnost. i Primenen. 17, 342-349, 1972.
• [33] K. Sato : Lévy processes and infinitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999.