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 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 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 Maximum principles for nonlocal parabolic Waldenfels operators

2019 ◽  
Vol 09 (03) ◽  
pp. 1950015 ◽  
Author(s):  
Qiao Huang ◽  
Jinqiao Duan ◽  
Jiang-Lun Wu
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Stochastic Differential Equations ◽  
Markov Processes ◽  
Parabolic Equations ◽  
Lévy Processes ◽  
Stochastic Dynamics ◽  
Levy Processes ◽  
Maximum Principles

As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for ‘parabolic’ equations with nonlocal Waldenfels operators. Applications in stochastic differential equations with [Formula: see text]-stable Lévy processes are presented to illustrate the maximum principles.

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

2012 ◽  
Vol 49 (3) ◽  
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 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.

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