Deep Factorisation of the Stable Process III: the View from Radial Excursion Theory and the Point of Closest Reach

We compute explicitly the distribution of the point of closest reach to the origin in the path of any d-dimensional isotropic stable process, with d ≥ 2. Moreover, we develop a new radial excursion theory, from which we push the classical Blumenthal–Getoor–Ray identities for first entry/exit into a ball (cf. Blumenthal et al. Trans. Amer. Math. Soc., 99, 540–554 1961) into the more complex setting of n-tuple laws for overshoots and undershoots. We identify explicitly the stationary distribution of any d-dimensional isotropic stable process when reflected in its running radial supremum. Finally, for such processes, and as consequence of some of the analysis of the aforesaid, we provide a representation of the Wiener–Hopf factorisation of the MAP that underlies the stable process through the Lamperti–Kiu transform. Our analysis continues in the spirit of Kyprianou (Ann. Appl. Probab., 20(2), 522–564 2010) and Kyprianou et al. (2015) in that our methodology is largely based around treating stable processes as self-similar Markov processes and, accordingly, taking advantage of their Lamperti-Kiu decomposition.

Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

Inspired by the works of Landriault et al. (2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. (2011), (2014).

RETRACTED ARTICLE: Fixed point theorems and explicit estimates for convergence rates of continuous time Markov chains

In this paper we give Banach fixed point theorems and explicit estimates on the rates of convergence of the transition function to the stationary distribution for a class of exponential ergodic Markov chains. Our results are different from earlier estimates using coupling theory, and from estimates using stochastically monotone one. Our estimates show a noticeable improvement on existing results if Markov chains contain instantaneous states or nonconservative states. The proof uses existing results of discrete time Markov chains together with h-skeleton. At last, we apply this result, Ray-Knight compactification and Itô excursion theory to two examples: a class of singular Markov chains and Kolmogorov matrix.

Walk in the history of the city and its environs

Having a look at the tourist space as a cultural specialist, the author drew attention to the fact that the closest to the modern man is a city environment he contacts and sometimes encounters in everyday life and on holidays. And every time whether he wants it or not, it opens in a dif erent way. One way of getting to know the world has long been a walking tour. It’s not just a walk hand in hand with a pleasant man or hasty movement to the right place, but namely the tour, in which a knowledgeable person with a soulful voice will speak about the past and present of the city and its surroundings, as if it is about your life and the people close to you. Turning to the beginning of the twentieth century, the experience of scientists-excursion specialists we today can learn a lot to improve the process of building up a tour, and most importantly the transmission of knowledge about the world in which we live. Well-known names of the excursion theory founders to professionals are I. Grevs, N. Antsiferov, N. Geynike and others. They are given in the context of ref ection on the historical development of walking tours, which haven’t lost their value and attract both creators and consumers of tour services.