Dyadic approximation of double integrals with respect to symmetric stable processes

Terry R. McConnell; Murad S. Taqqu

doi:10.1016/0304-4149(86)90009-8

Dyadic approximation of double integrals with respect to symmetric stable processes

Terry R. McConnell, Murad S. Taqqu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index α in [1, 2). In their approach the double integral is defined as an iterated Itô-type integral. We show here that it can also be defined as the limit of integrals of step functions and that the two approaches are equivalent. For many purposes this result reduces the study of double integrals to that of quadratic forms in independent stable random variables.

Original language	English (US)
Pages (from-to)	323-331
Number of pages	9
Journal	Stochastic Processes and their Applications
Volume	22
Issue number	2
DOIs	https://doi.org/10.1016/0304-4149(86)90009-8
State	Published - Jul 1986

Keywords

multiple Wiener integral
p-summing maps
radonifying maps
random quadratic forms
stochastic integration
symmetric stable processes

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Applied Mathematics

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10.1016/0304-4149(86)90009-8

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title = "Dyadic approximation of double integrals with respect to symmetric stable processes",

abstract = "Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index α in [1, 2). In their approach the double integral is defined as an iterated It{\^o}-type integral. We show here that it can also be defined as the limit of integrals of step functions and that the two approaches are equivalent. For many purposes this result reduces the study of double integrals to that of quadratic forms in independent stable random variables.",

keywords = "multiple Wiener integral, p-summing maps, radonifying maps, random quadratic forms, stochastic integration, symmetric stable processes",

author = "McConnell, {Terry R.} and Taqqu, {Murad S.}",

note = "Funding Information: * The first author is supported in part by NSF grant DMS 85-03775. ** The second author is supported in part by NSF grant ECS-84-08524.",

year = "1986",

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T1 - Dyadic approximation of double integrals with respect to symmetric stable processes

AU - McConnell, Terry R.

AU - Taqqu, Murad S.

N1 - Funding Information: * The first author is supported in part by NSF grant DMS 85-03775. ** The second author is supported in part by NSF grant ECS-84-08524.

PY - 1986/7

Y1 - 1986/7

N2 - Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index α in [1, 2). In their approach the double integral is defined as an iterated Itô-type integral. We show here that it can also be defined as the limit of integrals of step functions and that the two approaches are equivalent. For many purposes this result reduces the study of double integrals to that of quadratic forms in independent stable random variables.

AB - Recently, Rosinski and Woyczynski have given necessary and sufficient conditions for the existence of the double integral with respect to a symmetric stable process of index α in [1, 2). In their approach the double integral is defined as an iterated Itô-type integral. We show here that it can also be defined as the limit of integrals of step functions and that the two approaches are equivalent. For many purposes this result reduces the study of double integrals to that of quadratic forms in independent stable random variables.

KW - multiple Wiener integral

KW - p-summing maps

KW - radonifying maps

KW - random quadratic forms

KW - stochastic integration

KW - symmetric stable processes

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DO - 10.1016/0304-4149(86)90009-8

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AN - SCOPUS:46149128191

SN - 0304-4149

VL - 22

SP - 323

EP - 331

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 2

ER -

Dyadic approximation of double integrals with respect to symmetric stable processes

Abstract

Keywords

ASJC Scopus subject areas

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