Infinite divisibility for stochastic processes and time change
type: Item
instance of: scholarly article
title: Infinite divisibility for stochastic processes and time change
zbMATH Open document ID: 1111.60028
author: M. Maejima
author: Ken-Iti Sato
author: Ole E. Barndorff-Nielsen
published in: Journal of Theoretical Probability
publication date: 2007-02-14T00:00:00Z
review text: The authors present some general results on infinite divisibility, self-decomposability and the class \(L_m\) (the class of measures \(\mu\) such that for any \(c\in (0,1)\) there is a distribution \(\rho^{(c)}\) such that \(\hat{\mu}(z)=\hat{\mu}(cz)\hat{\rho^{(c)}}(z)\), \(z\in \mathbb{R}^d\), and \(\hat{\mu}(z)=\hat{\mu}(cz)\hat{\rho^{(c)}}(z)\) also holds with some \(\rho^{(c)}\in L_{m-1}\)). Several relations between various concepts and the properties of such processes are given. Also, a new concept of temporal self-decomposability is introduced: the process is called temporal self-decomposable if for each \(c\in (0,1)\) there exist independent processes \(X^{(c)}\) and \(U^{(c)}\) such that \(X\overset{d}{=} X^{(c)}+U^{(c)},\) and \(X^{(c)}=\{X^{(c)}_t:\,\,t\geq 0\}\overset{d}{=} \{X_{ct}\,\, t\geq 0\}\). The class of temporal self-decomposable processes is larger than the class of Lévy processes. On the other hand, under some restriction temporary self-decomposable processes are infinitely divisible. The notion of additive processes is also between the notion of Lévy processes and the notion of infinitely divisible processes. It is shown that an additive process in not always temporary self-decomposable, and vice versa. Various examples are given. Further, the time-change of stochastic processes is discussed, in particular, the authors present the results on the inheritance of infinite divisibility under time change when the base process (i.e. the process which is to be time-changed) is a Lévy process.
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 60G18
zbMATH DE Number: 5126776
zbMATH Keywords: self-decomposability
zbMATH Keywords: temporal self-decomposability
zbMATH Keywords: class \(L_m\)-property
zbMATH Keywords: time change
zbMATH Keywords: chronometer
reviewed by: V. Knopova
MaRDI profile type: Publication
full work available at URL: s10959-006-0020-7
OpenAlex ID: W2045819438
cites work: On certain self-decomposable self-similar processes with independent increments
cites work: Multivariate subordination, self-decomposability and stability
cites work: Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics. (With discussion)
cites work: Impact of jumps on returns and realised variances: econometric analysis of time-deformed Lévy processes
cites work: Power Variation and Time Change
cites work: Change of Time and Change of Measure
cites work: Q4888858
cites work: Q4941947
cites work: Diffusion Equation and Stochastic Processes
cites work: Q3229718
cites work: Stochastic Volatility for Lévy Processes
cites work: Q4441722
cites work: Financial Modelling with Jump Processes
cites work: Q2738734
cites work: Q2782356
cites work: Q3149666
cites work: Time changes for Lévy processes
cites work: Stochastic volatility, jumps and hidden time changes
cites work: Self-similar processes with independent increments associated with Lévy and Bessel processes.
cites work: Log-fractional stable processes
cites work: Semi-Lévy processes, semi-selfsimilar additive processes, and semi-stationary Ornstein-Uhlen\-beck type processes
cites work: Distributions of selfsimilar and semi-selfsimilar processes with independent increments
cites work: Some multivariate infinitely divisible distributions and their projections
cites work: Q3213952
cites work: Cone-parameter convolution semigroups and their subordination
cites work: Q4461460
cites work: Q4301585
cites work: Self-similar processes with independent increments
cites work: Q4225858
cites work: Q4937701
cites work: Stochastic integrals in additive processes and application to semi-Lévy processes
DOI: 10.1007/S10959-006-0020-7
description: scientific article; zbMATH DE number 5126776