The asymptotic behaviour of (Yn, n e N) is of fundamental importance in probability theory. Indeed, if the Xj have common mean fi and variance a, then by taking each an = n/u and b„ = n a, the celebrated Central Limit Theorem tells us that (Yn) converges in distribution to a standard normal. The class of random variables which appear as limits in distribution of (Yn) for more general (an) and (b„) are called the stable random variables and they have a number of elegant properties, such as the preservation of their distribution functions under convolution. A stochastic process is said to be stable if all its finite-dimensional distributions are themselves stable and it is the systematic study of such processes which is the aim of the present volume.

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

A Primer on Feller Semigroups and Feller Processes- Feller Generators and Symbols- Construction of Feller Processes- Transformations of Feller Processes- Sample Path Properties- Global Properties- Approximation- Open Problems- References- Index

Lévy-type processes : construction, approximation and sample path properties

https://bib.irb.hr/datoteka/607011.N._Sandric_Long-time_behavior_of_stable-like_processes.pdf

Long-time behavior of stable-like processes

Nous etudions l’ergodicite exponentielle d’une large classe de processus stochastiques dont la dynamique est regie par un bruit de Levy a sauts purs. Dans la premiere partie de l’article, nous nous concentrons sur les solutions d’equations differentielles stochastiques (SDEs) dont les derives satisfont aux conditions generales de type Lyapunov. En appliquant des techniques qui combinent des couplages, des distances de L1-Wasserstein et des fonctions de Lyapunov correctement construites, nous montrons une convergence exponentielle des solutions de telles SDE vers leurs distributions stationnaires, a la fois sous la distance de variation totale et de Wasserstein. La deuxieme partie de l’article est consacree aux SDEs de type McKean–Vlasov avec des derives dependant de la distribution. Nous prouvons un resultat de propagation de chaos uniformement en temps, en fournissant des bornes quantitatives sur le taux de convergence des systemes de particules en interaction avec le bruit de Levy vers le SDE deMcKean–Vlasov correspondant. Ensuite, en etendant nos techniques dans la premiere partie de l’article, nous obtenons des resultats sur l’ergodicite exponentielle des solutions de SDE de McKean–Vlasov, dans des conditions generales sur la derive et le bruit.

Exponential ergodicity for SDEs and McKean–Vlasov processes with Lévy noise

This work focuses on a class of jump-diffusions with state-dependent switching. First, compared with the existing results in the literature, in our model, the characteristic measure is allowed to be σ-finite. The existence and uniqueness of the underlying process are obtained by representing the switching component as a stochastic integral with respect to a Poisson random measure and by using a successive approximation method. Then, the Feller property is proved by means of introducing auxiliary processes and by making use of Radon-Nikodym derivatives. Furthermore, the irreducibility and all compact sets being petite are demonstrated. Based on these results, the uniform ergodicity is established under a general Lyapunov condition. Finally, easily verifiable conditions for uniform ergodicity are established when the jump-diffusions are linearizable with respect to the variable x (the state variable corresponding to the jump-diffusion component) in a neighborhood of the infinity, and some examples are presented to illustrate the results.

Jump-diffusions with state-dependent switching: existence and uniqueness, Feller property, linearization, and uniform ergodicity

Criteria for ergodicity of Lévy type operators in dimension one

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Criteria for ergodicity of Lévy type operators in dimension one