Recurrent Extensions of Real-Valued Self-Similar Markov Processes

H. Pantí, J. C. Pardo, V. M. Rivero

Resultado de la investigación: Contribución a una revistaArtículo

Resumen

Let X = (Xt,t ≥ 0) be a self-similar Markov process taking values in ℝ such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (Electron. Commun. Probab. 11, 230–241 2006) and Rivero (Bernoulli 11, 471–509 2005, 13, 1053–1070 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable Lévy process which to the best of our knowledge has not been studied before.

Idioma originalInglés
PublicaciónPotential Analysis
DOI
EstadoPublicada - 1 ene 2019

Huella dactilar

Self-similar Processes
Markov Process
Markov Additive Process
Stable Process
Trap
Bernoulli
Existence and Uniqueness
Electron
Necessary Conditions
Sufficient Conditions

Citar esto

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title = "Recurrent Extensions of Real-Valued Self-Similar Markov Processes",
abstract = "Let X = (Xt,t ≥ 0) be a self-similar Markov process taking values in ℝ such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (Electron. Commun. Probab. 11, 230–241 2006) and Rivero (Bernoulli 11, 471–509 2005, 13, 1053–1070 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable L{\'e}vy process which to the best of our knowledge has not been studied before.",
keywords = "Exponential functional, Lamperti–Kiu representation, Markov additive processes, Real self-similar Markov processes, Stable processes",
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Recurrent Extensions of Real-Valued Self-Similar Markov Processes. / Pantí, H.; Pardo, J. C.; Rivero, V. M.

En: Potential Analysis, 01.01.2019.

Resultado de la investigación: Contribución a una revistaArtículo

TY - JOUR

T1 - Recurrent Extensions of Real-Valued Self-Similar Markov Processes

AU - Pantí, H.

AU - Pardo, J. C.

AU - Rivero, V. M.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let X = (Xt,t ≥ 0) be a self-similar Markov process taking values in ℝ such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (Electron. Commun. Probab. 11, 230–241 2006) and Rivero (Bernoulli 11, 471–509 2005, 13, 1053–1070 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable Lévy process which to the best of our knowledge has not been studied before.

AB - Let X = (Xt,t ≥ 0) be a self-similar Markov process taking values in ℝ such that the state 0 is a trap. In this paper, we present a necessary and sufficient condition for the existence of a self-similar recurrent extension of X that leaves 0 continuously. The condition is expressed in terms of the associated Markov additive process via the Lamperti-Kiu representation. Our results extend those of Fitzsimmons (Electron. Commun. Probab. 11, 230–241 2006) and Rivero (Bernoulli 11, 471–509 2005, 13, 1053–1070 2007) where the existence and uniqueness of a recurrent extension for positive self similar Markov processes were treated. In particular, we describe the recurrent extension of a stable Lévy process which to the best of our knowledge has not been studied before.

KW - Exponential functional

KW - Lamperti–Kiu representation

KW - Markov additive processes

KW - Real self-similar Markov processes

KW - Stable processes

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JF - Potential Analysis

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