Cari colleghi

scusandomi per eventuali messaggi multipli, vi mando le informazioni sui prossimi seminari online del Gruppo UMI Prisma (lunedì 4 aprile), con i contributi di Enrico Scalas e Giacomo Ascione:

* April 4, 2022, 16:00-17:00 (CET): Enrico Scalas

TITLE:

Point processes and time change: A fractional non-homogeneous Poisson process and its functional limits

ABSTRACT:

A fractional nonhomogeneous Poisson process was introduced by a time change of the nonhomogeneous Poisson process with the inverse α-stable subordinator. A similar definition is proposed for the (nonhomogeneous) fractional compound Poisson process. Both finite-dimensional and functional limit theorems are presented for the fractional nonhomogeneous Poisson process and the fractional compound Poisson process. The results are derived by using martingale methods, regular variation properties and Anscombe’s theorem. Some of the limiting results are verified in a Monte Carlo simulation. Papers: [1] Nikolai Leonenko, Enrico Scalas and Mailan Trinh, The fractional non-homogeneous Poisson process. Statistics and Probability Letters, 120, 2017, pp. 147-156. DOI: http://dx.doi.org/10.1016/j.spl.2016.09.024 https://arxiv.org/abs/1601.03965 [2] Nikolai Leonenko, Enrico Scalas and Mailan Trinh, Limit theorems for the fractional nonhomogeneous Poisson process, Journal of Applied Probability , 56:1, 2019 , pp. 246 - 264. DOI: https://doi.org/10.1017/jpr.2019.16 https://arxiv.org/abs/1711.08768 This is joint work with Nikolai Leonenko and Mailan Trinh.

* April 4, 2022, 17:00-18:00 (CET): Giacomo Ascione

TITLE:

Spectral methods for time-changed birth-death processes

ABSTRACT:

In this talk we focus on a class of semi-Markov birth-death processes obtained by means of a time-change of some standard birth-death process. Precisely, we consider as parent processes the immigration-death process and the Meixner process, whose stationary distributions are respectively the Poisson and the Pascal distributions. Exploiting, on one hand, the properties of the Charlier and Meixner polynomials (in particular, the self-duality property), while, on the other, characterizing the eigenfunctions of some non-local operators by means of the Laplace transform of an inverse subordinator, we are able to explicitly express the spectral decomposition of the transition probability function of the aforementioned processes. The latter expression is then used to prove existence and uniqueness of strong solutions for a class of time-nonlocal Cauchy problems in a suitable Banach sequence space and the probabilistic interpretation of such equations as some sort of non-local backward/forward Kolmogorov equations. Finally, a comparison with the time-changed diffusion case is carried out by referring to the spectral decomposition of the probability density function of time-changed Pearson diffusions. The latter argument hints at the possibility of applying this kind of spectral methods to a wider range of problems. This is the result of joint work with Nikolai Leonenko from Cardiff University and Enrica Pirozzi from University of Naples.

Grazie per l'attenzione, Domenico Marinucci

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