# Victor Manuel Rivero Mercado - Representations of the Wiener-Hopf factorization for Markov additive processes

Dates: | 21 February 2024 |
---|---|

Times: | 16:00 - 17:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Victor Manuel Rivero Mercado (Center of Research in Mathematics in Guanajuato, Mexico) will speak at the Probability seminar.

A keystone in the fluctuation theory of Levy processes is the Wiener-Hopf factorization of the characteristic exponent, and most of the research on this topic involves consequences or facts about it. It contains suitable information to describe the so-called upward and ladder height process associated with a Levy process. These processes lead to a deep understanding of the running supremum and infimum of the path, and in principle, have a tractable structure as their paths are non-decreasing. Recently there has been a renewed interest in the class of Markov additive processes, due in particular to its connection to self-similar Markov processes and the richness of its structure. These processes can be thought of as Levy processes modulated by an auxiliary Markov process, as they are composed of an additive and a driving Markovian component. The additive component bears various nice properties of Levy processes, for instance, stationarity and homogeneity of their increments, but conditionally to the driving component. When the driving component is a Markov chain with a finite state space, we can think of the driving part as a concatenation of Levy processes, and hence we get easily convinced that many results for Levy processes should have a translation into this setting. In particular, it has been proved that the Wiener-Hopf factorization holds true and that several of its representations can be extended to this setting. A particularly interesting one was obtained by Vigon, it relates the Levy measures of the Levy process and those of their upward and downward ladder height processes, the so-called equations amicales. This has been recently proved to hold in the Markov additive context by Doering, Trotter and Watson, under the assumption of the state space of the driving process being finite. In this talk, I will describe the form the Wiener-Hopf factorization takes in the more general setting where the driving process has a general state space, and prove that the equations amicales hold also true in this general setting. The proof we will provide of this result uses some general principles of Markov processes, for instance, duality, resolvents and infinitesimal generators. This talk is based on a work in progress in collaboration with Andreas Kyprianou and Mehar Motala.