Discretization of the Lamperti representation of a positive self-similar Markov process

Jevgenijs Ivanovs; Jakob D. Thøstesen

doi:10.1016/j.spa.2021.03.013

Discretization of the Lamperti representation of a positive self-similar Markov process

^*Corresponding author for this work

Department of Mathematics

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Abstract

This paper considers discretization of the Lévy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given Lévy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar Lévy processes conditioned to stay positive.

Original language	English
Journal	Stochastic Processes and Their Applications
Volume	137
Pages (from-to)	200-221
Number of pages	22
ISSN	0304-4149
DOIs	https://doi.org/10.1016/j.spa.2021.03.013
Publication status	Published - Jul 2021

Keywords

Exponential functional
Lamperti representation
Positive self-similar Markov process
Small time behavior
Stable Lévy process conditioned to stay positive

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10.1016/j.spa.2021.03.013

https://arxiv.org/pdf/2006.09115.pdf

Cite this

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title = "Discretization of the Lamperti representation of a positive self-similar Markov process",

abstract = "This paper considers discretization of the L{\'e}vy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given L{\'e}vy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar L{\'e}vy processes conditioned to stay positive.",

keywords = "Exponential functional, Lamperti representation, Positive self-similar Markov process, Small time behavior, Stable L{\'e}vy process conditioned to stay positive",

author = "Jevgenijs Ivanovs and Th{\o}stesen, {Jakob D.}",

note = "Publisher Copyright: {\textcopyright} 2021 Elsevier B.V.",

year = "2021",

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AU - Ivanovs, Jevgenijs

AU - Thøstesen, Jakob D.

PY - 2021/7

Y1 - 2021/7

N2 - This paper considers discretization of the Lévy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given Lévy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar Lévy processes conditioned to stay positive.

AB - This paper considers discretization of the Lévy process appearing in the Lamperti representation of a strictly positive self-similar Markov process. Limit theorems for the resulting approximation are established under some regularity assumptions on the given Lévy process. Additionally, the scaling limit of a positive self-similar Markov process at small times is provided. Finally, we present an application to simulation of self-similar Lévy processes conditioned to stay positive.

KW - Exponential functional

KW - Lamperti representation

KW - Positive self-similar Markov process

KW - Small time behavior

KW - Stable Lévy process conditioned to stay positive

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U2 - 10.1016/j.spa.2021.03.013

DO - 10.1016/j.spa.2021.03.013

M3 - Journal article

AN - SCOPUS:85104087100

SN - 0304-4149

VL - 137

SP - 200

EP - 221

JO - Stochastic Processes and Their Applications

JF - Stochastic Processes and Their Applications

ER -

Discretization of the Lamperti representation of a positive self-similar Markov process

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Keywords

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Local Behavior and Graphical Models for Stochastic Processes