scholarly journals Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes

2006 ◽  
Vol 34 (3) ◽  
pp. 1012-1034 ◽  
Author(s):  
M. E. Caballero ◽  
L. Chaumont
Keyword(s):  
Weak Convergence ◽  
Markov Processes ◽  
Lévy Processes ◽  
Levy Processes ◽  
Self Similar

scholarly journals Double hypergeometric Lévy processes and self-similarity

2021 ◽  
Vol 58 (1) ◽  
pp. 254-273
Author(s):  
Andreas E. Kyprianou ◽  
Juan Carlos Pardo ◽  
Matija Vidmar
Keyword(s):  
Closed Form ◽  
Markov Processes ◽  
Lévy Processes ◽  
Levy Processes ◽  
Stable Processes ◽  
Self Similarity ◽  
New Family ◽  
Self Similar

AbstractMotivated by a recent paper (Budd (2018)), where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener–Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

scholarly journals CLASSICAL MARKOV PROCESSES FROM QUANTUM LÉVY PROCESSES

1999 ◽  
Vol 02 (01) ◽  
pp. 105-129 ◽  
Author(s):  
UWE FRANZ
Keyword(s):  
Markov Process ◽  
Hopf Algebra ◽  
Markov Processes ◽  
Lévy Processes ◽  
Sufficient Conditions ◽  
Markov Property ◽  
Vacuum State ◽  
Levy Processes ◽  
Quantum Process ◽  
Quantum Markov Processes

We show how classical Markov processes can be obtained from quantum Lévy processes. It is shown that quantum Lévy processes are quantum Markov processes, and sufficient conditions for restrictions to subalgebras to remain quantum Markov processes are given. A classical Markov process (which has the same time-ordered moments as the quantum process in the vacuum state) exists whenever we can restrict to a commutative subalgebra without losing the quantum Markov property.8 Several examples, including the Azéma martingale, with explicit calculations are presented. In particular, the action of the generator of the classical Markov processes on polynomials or their moments are calculated using Hopf algebra duality.

scholarly journals On transient Markov processes of Ornstein-Uhlenbeck type

1998 ◽  
Vol 149 ◽  
pp. 19-32 ◽  
Author(s):  
Kouji Yamamuro
Keyword(s):  
Markov Processes ◽  
Lévy Processes ◽  
Transition Probabilities ◽  
Levy Processes ◽  
Exit Times ◽  
Linear Drift ◽  
Lévy Measures ◽  
Drift Terms

Abstract.For Hunt processes on Rd, strong and weak transience is defined by finiteness and infiniteness, respectively, of the expected last exit times from closed balls. Under some condition, which is satisfied by Lévy processes and Ornstein-Uhlenbeck type processes, this definition is expressed in terms of the transition probabilities. A criterion is given for strong and weak transience of Ornstein-Uhlenbeck type processes on Rd, using their Lévy measures and coefficient matrices of linear drift terms. An example is discussed.

scholarly journals Apparent multifractality of self-similar Lévy processes

Nonlinearity ◽  
2017 ◽  
Vol 30 (7) ◽  
pp. 2592-2611
Author(s):  
Marco Zamparo
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Self Similar

scholarly journals A New Proof of the Wiener-Hopf Factorization via Basu's Theorem

2012 ◽  
Vol 49 (03) ◽  
pp. 876-882
Author(s):  
Brian Fralix ◽  
Colin Gallagher
Keyword(s):  
Random Walks ◽  
Markov Processes ◽  
Lévy Processes ◽  
Levy Processes ◽  
Specific Class ◽  
Piecewise Deterministic Markov Processes

We illustrate how Basu's theorem can be used to derive the spatial version of the Wiener-Hopf factorization for a specific class of piecewise-deterministic Markov processes. The classical factorization results for both random walks and Lévy processes follow immediately from our result. The approach is particularly elegant when used to establish the factorization for spectrally one-sided Lévy processes.

scholarly journals Stationary and self-similar processes driven by Lévy processes

1999 ◽  
Vol 84 (2) ◽  
pp. 357-369 ◽  
Author(s):  
Ole E Barndorff-Nielsen ◽  
Victor Pérez-Abreu
Keyword(s):  
Lévy Processes ◽  
Levy Processes ◽  
Self Similar

scholarly journals Conditioned stable Lévy processes and the Lamperti representation

2006 ◽  
Vol 43 (04) ◽  
pp. 967-983 ◽  
Author(s):  
M. E. Caballero ◽  
L. Chaumont
Keyword(s):  
Ruin Probability ◽  
Lévy Process ◽  
Lévy Processes ◽  
First Passage Time ◽  
Passage Time ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Lévy Process ◽  
Self Similar ◽  
Positive Half

By variously killing a stable Lévy process when it leaves the positive half-line, conditioning it to stay positive, and conditioning it to hit 0 continuously, we obtain three different, positive, self-similar Markov processes which illustrate the three classes described by Lamperti (1972). For each of these processes, we explicitly compute the infinitesimal generator and from this deduce the characteristics of the underlying Lévy process in the Lamperti representation. The proof of this result bears on the behaviour at time 0 of stable Lévy processes before their first passage time across level 0, which we describe here. As an application, for a certain class of Lévy processes we give the law of the minimum before an independent exponential time. This provides the explicit form of the spatial Wiener-Hopf factor at a particular point and the value of the ruin probability for this class of Lévy processes.

Sign in / Sign up

Export Citation Format

Share Document