Mimicking self-similar processes

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4 Citations (Scopus)

Abstract

We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales as a generalization of Brownian motion, martingales of the squared Bessel process, stable Lévy processes as well as an example of an artificial process having the marginals of tκVtκV for some symmetric random variable VV. At the end, we see how we can mimic certain Brownian martingales which are non-Markovian.
Original languageEnglish
Pages (from-to)1341-1360
Number of pages20
JournalBernoulli
Volume21
Issue number3
DOIs
Publication statusPublished - 2015

Keywords

  • Lévy processes
  • martingales with given marginals
  • self-similar

Cite this

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title = "Mimicking self-similar processes",
abstract = "We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales as a generalization of Brownian motion, martingales of the squared Bessel process, stable L{\'e}vy processes as well as an example of an artificial process having the marginals of tκVtκV for some symmetric random variable VV. At the end, we see how we can mimic certain Brownian martingales which are non-Markovian.",
keywords = "L{\'e}vy processes, martingales with given marginals, self-similar",
author = "Fan, {Jie Yen} and Kais Hamza and Klebaner, {Fima C}",
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pages = "1341--1360",
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Mimicking self-similar processes. / Fan, Jie Yen; Hamza, Kais; Klebaner, Fima C.

In: Bernoulli, Vol. 21, No. 3, 2015, p. 1341-1360.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

T1 - Mimicking self-similar processes

AU - Fan, Jie Yen

AU - Hamza, Kais

AU - Klebaner, Fima C

PY - 2015

Y1 - 2015

N2 - We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales as a generalization of Brownian motion, martingales of the squared Bessel process, stable Lévy processes as well as an example of an artificial process having the marginals of tκVtκV for some symmetric random variable VV. At the end, we see how we can mimic certain Brownian martingales which are non-Markovian.

AB - We construct a family of self-similar Markov martingales with given marginal distributions. This construction uses the self-similarity and Markov property of a reference process to produce a family of Markov processes that possess the same marginal distributions as the original process. The resulting processes are also self-similar with the same exponent as the original process. They can be chosen to be martingales under certain conditions. In this paper, we present two approaches to this construction, the transition-randomising approach and the time-change approach. We then compute the infinitesimal generators and obtain some path properties of the resulting processes. We also give some examples, including continuous Gaussian martingales as a generalization of Brownian motion, martingales of the squared Bessel process, stable Lévy processes as well as an example of an artificial process having the marginals of tκVtκV for some symmetric random variable VV. At the end, we see how we can mimic certain Brownian martingales which are non-Markovian.

KW - Lévy processes

KW - martingales with given marginals

KW - self-similar

UR - https://projecteuclid.org/download/pdfview_1/euclid.bj/1432732022

U2 - 10.3150/13-BEJ588

DO - 10.3150/13-BEJ588

M3 - Article

VL - 21

SP - 1341

EP - 1360

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 3

ER -