Stable processes conditioned to avoid an interval

Leif Döring, Andreas E. Kyprianou, Philip Weissmann

Research output: Contribution to journalArticle

Abstract

Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function. In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for α≥1 go back to classical work of Blumenthal et al. and Port from the 1960s. For α<1, we appeal to recent results centered around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all α∈(0,2). With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob h-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob h-transform, we verify that the conditioned process is transient.

Original languageEnglish
JournalStochastic Processes and their Applications
Early online date19 Feb 2019
DOIs
Publication statusE-pub ahead of print - 19 Feb 2019

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics

Cite this

Stable processes conditioned to avoid an interval. / Döring, Leif; Kyprianou, Andreas E.; Weissmann, Philip.

In: Stochastic Processes and their Applications, 19.02.2019.

Research output: Contribution to journalArticle

@article{8b46b1226bbd4e0ab6ed06f7bfedbe64,
title = "Stable processes conditioned to avoid an interval",
abstract = "Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function. In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for α≥1 go back to classical work of Blumenthal et al. and Port from the 1960s. For α<1, we appeal to recent results centered around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all α∈(0,2). With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob h-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob h-transform, we verify that the conditioned process is transient.",
author = "Leif D{\"o}ring and Kyprianou, {Andreas E.} and Philip Weissmann",
year = "2019",
month = "2",
day = "19",
doi = "10.1016/j.spa.2019.01.004",
language = "English",
journal = "Stochastic Processes and their Applications",
issn = "0304-4149",
publisher = "Elsevier",

}

TY - JOUR

T1 - Stable processes conditioned to avoid an interval

AU - Döring, Leif

AU - Kyprianou, Andreas E.

AU - Weissmann, Philip

PY - 2019/2/19

Y1 - 2019/2/19

N2 - Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function. In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for α≥1 go back to classical work of Blumenthal et al. and Port from the 1960s. For α<1, we appeal to recent results centered around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all α∈(0,2). With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob h-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob h-transform, we verify that the conditioned process is transient.

AB - Conditioning Markov processes to avoid a domain is a classical problem that has been studied in many settings. Ingredients for standard arguments involve the leading order tail asymptotics of the distribution of the first hitting time of the domain of interest and its relation to an underlying harmonic function. In the present article we condition stable processes to avoid intervals. The required tail asymptotics in the stable setting for α≥1 go back to classical work of Blumenthal et al. and Port from the 1960s. For α<1, we appeal to recent results centered around the so-called deep factorisation of the stable process to compute hitting probabilities and, moreover, to identify the associated harmonic functions for all α∈(0,2). With these in hand, we thus prove that conditioning to avoid an interval is possible in the classical sense and that the resulting process is a Doob h-transform of the stable process killed on entering the aforesaid interval. Appealing to the representation of the conditioned process as a Doob h-transform, we verify that the conditioned process is transient.

UR - http://www.scopus.com/inward/record.url?scp=85062283018&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2019.01.004

DO - 10.1016/j.spa.2019.01.004

M3 - Article

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

ER -