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2006 Additive processes and stochastic integrals
Ken-Iti Sato
Illinois J. Math. 50(1-4): 825-851 (2006). DOI: 10.1215/ijm/1258059494

Abstract

Stochastic integrals of nonrandom $(l\times d)$-matrix-valued functions or nonrandom real-valued functions with respect to an additive process $X$ on $\mathbb{R}^d$ are studied. Here an additive process means a stochastic process with independent increments, stochastically continuous, starting at the origin, and having cadlag paths. A necessary and sufficient condition for local integrability of matrix-valued functions is given in terms of the Lévy--Khintchine triplets of a factoring of $X$. For real-valued functions explicit expressions of the condition are presented for all semistable Lévy processes on $\mathbb{R}^d$ and some selfsimilar additive processes. In the last part of the paper, existence conditions for improper stochastic integrals $\int_0^{\infty-}f(s)dX_s$ and their extensions are given; the cases where $f(s)\asymp s^{\beta} e^{-cs^{\alpha}}$ and where $f(s)$ is such that $s=\int_{f(s)}^{\infty} u^{-2} e^{-u} du$ are analyzed.

Citation

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Ken-Iti Sato. "Additive processes and stochastic integrals." Illinois J. Math. 50 (1-4) 825 - 851, 2006. https://doi.org/10.1215/ijm/1258059494

Information

Published: 2006
First available in Project Euclid: 12 November 2009

zbMATH: 1103.60051
MathSciNet: MR2247848
Digital Object Identifier: 10.1215/ijm/1258059494

Subjects:
Primary: 60G51
Secondary: 60E07, 60H05

Rights: Copyright © 2006 University of Illinois at Urbana-Champaign

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Vol.50 • No. 1-4 • 2006
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