Lévy processes conditioned to stay in a half-space with applications to directional extremes

Jevgenijs Ivanovs; Jakob Dalsgaard Thøstesen

doi:10.15559/22-VMSTA217

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Jevgenijs Ivanovs, Jakob Dalsgaard Thøstesen^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

Abstract

This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

Original language	English
Journal	Modern Stochastics: Theory and Applications
Volume	10
Issue	1
Pages (from-to)	59-75
Number of pages	17
ISSN	2351-6046
DOIs	https://doi.org/10.15559/22-VMSTA217
Publication status	Published - Jan 2023

Keywords

Conditioning to stay positive
Sparre Andersen identity
directional extremes
exchangeability
local behavior

Access to Document

10.15559/22-VMSTA217

1 Ph.D. thesis

Local Behavior and Graphical Models for Stochastic Processes
Thøstesen, J. D., Oct 2023, Århus Universitet.
Research output: Book/anthology/dissertation/report › Ph.D. thesis

Cite this

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title = "L{\'e}vy processes conditioned to stay in a half-space with applications to directional extremes",

abstract = "This paper provides a multivariate extension of Bertoin{\textquoteright}s pathwise construction of a L{\'e}vy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a L{\'e}vy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.",

keywords = "Conditioning to stay positive, Sparre Andersen identity, directional extremes, exchangeability, local behavior",

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

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doi = "10.15559/22-VMSTA217",

language = "English",

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Lévy processes conditioned to stay in a half-space with applications to directional extremes. / Ivanovs, Jevgenijs; Thøstesen, Jakob Dalsgaard.
In: Modern Stochastics: Theory and Applications, Vol. 10, No. 1, 01.2023, p. 59-75.

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

TY - JOUR

T1 - Lévy processes conditioned to stay in a half-space with applications to directional extremes

AU - Ivanovs, Jevgenijs

AU - Thøstesen, Jakob Dalsgaard

PY - 2023/1

Y1 - 2023/1

N2 - This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

AB - This paper provides a multivariate extension of Bertoin’s pathwise construction of a Lévy process conditioned to stay positive or negative. Thus obtained processes conditioned to stay in half-spaces are closely related to the original process on a compact time interval seen from its directional extremal points. In the case of a correlated Brownian motion the law of the conditioned process is obtained by a linear transformation of a standard Brownian motion and an independent Bessel-3 process. Further motivation is provided by a limit theorem corresponding to zooming in on a Lévy process with a Brownian part at the point of its directional infimum. Applications to zooming in at the point furthest from the origin are envisaged.

KW - Conditioning to stay positive

KW - Sparre Andersen identity

KW - directional extremes

KW - exchangeability

KW - local behavior

U2 - 10.15559/22-VMSTA217

DO - 10.15559/22-VMSTA217

M3 - Journal article

SN - 2351-6046

VL - 10

SP - 59

EP - 75

JO - Modern Stochastics: Theory and Applications

JF - Modern Stochastics: Theory and Applications

IS - 1

ER -

Lévy processes conditioned to stay in a half-space with applications to directional extremes

Abstract

Keywords

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Research output

Local Behavior and Graphical Models for Stochastic Processes

Cite this