On the construction of Wiener integrals with respect to certain pseudo-Bessel processes

T. Funaki; Y. Hariya; F. Hirsch; M. Yor

doi:10.1016/j.spa.2006.05.001

On the construction of Wiener integrals with respect to certain pseudo-Bessel processes

T. Funaki, Y. Hariya, F. Hirsch^*, M. Yor

^*Corresponding author for this work

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

2 Citations (Scopus)

Abstract

In previous papers [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes Related Fields (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press)], the authors have shown that it is possible to define the Wiener-type integrals ∫₀¹ h (s) d over(R, -)_s, for every h ∈ L² ([0, 1], d s) and (over(R, -)_s) any centered Bessel process with dimension d > 1. In this paper, various conditions are stated, showing that such a construction is possible for a large class of processes indexed by two square integrable Brownian functionals. In particular, some of the results previously obtained for the Bessel processes are thus recovered, and in fact shown to extend to certain processes of the form sqrt(t) f (frac(R_t, sqrt(t))).

Original language	English
Pages (from-to)	1690-1711
Number of pages	22
Journal	Stochastic Processes and their Applications
Volume	116
Issue number	12
DOIs	https://doi.org/10.1016/j.spa.2006.05.001
Publication status	Published - 2006 Dec
Externally published	Yes

Keywords

Bessel processes
Gebelein's inequality
Hermite and Laguerre series expansions
Itô's representation theorem
Scaling property
Wiener integrals

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

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10.1016/j.spa.2006.05.001

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title = "On the construction of Wiener integrals with respect to certain pseudo-Bessel processes",

abstract = "In previous papers [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes Related Fields (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press)], the authors have shown that it is possible to define the Wiener-type integrals ∫01 h (s) d over(R, -)s, for every h ∈ L2 ([0, 1], d s) and (over(R, -)s) any centered Bessel process with dimension d > 1. In this paper, various conditions are stated, showing that such a construction is possible for a large class of processes indexed by two square integrable Brownian functionals. In particular, some of the results previously obtained for the Bessel processes are thus recovered, and in fact shown to extend to certain processes of the form sqrt(t) f (frac(Rt, sqrt(t))).",

keywords = "Bessel processes, Gebelein's inequality, Hermite and Laguerre series expansions, It{\^o}'s representation theorem, Scaling property, Wiener integrals",

author = "T. Funaki and Y. Hariya and F. Hirsch and M. Yor",

year = "2006",

month = dec,

doi = "10.1016/j.spa.2006.05.001",

language = "English",

volume = "116",

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journal = "Stochastic Processes and their Applications",

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T1 - On the construction of Wiener integrals with respect to certain pseudo-Bessel processes

AU - Funaki, T.

AU - Hariya, Y.

AU - Hirsch, F.

AU - Yor, M.

PY - 2006/12

Y1 - 2006/12

N2 - In previous papers [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes Related Fields (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press)], the authors have shown that it is possible to define the Wiener-type integrals ∫01 h (s) d over(R, -)s, for every h ∈ L2 ([0, 1], d s) and (over(R, -)s) any centered Bessel process with dimension d > 1. In this paper, various conditions are stated, showing that such a construction is possible for a large class of processes indexed by two square integrable Brownian functionals. In particular, some of the results previously obtained for the Bessel processes are thus recovered, and in fact shown to extend to certain processes of the form sqrt(t) f (frac(Rt, sqrt(t))).

AB - In previous papers [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered powers of Bessel processes, I, Markov Processes Related Fields (2006) (in press); T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes, II, Alea (2006) (in press)], the authors have shown that it is possible to define the Wiener-type integrals ∫01 h (s) d over(R, -)s, for every h ∈ L2 ([0, 1], d s) and (over(R, -)s) any centered Bessel process with dimension d > 1. In this paper, various conditions are stated, showing that such a construction is possible for a large class of processes indexed by two square integrable Brownian functionals. In particular, some of the results previously obtained for the Bessel processes are thus recovered, and in fact shown to extend to certain processes of the form sqrt(t) f (frac(Rt, sqrt(t))).

KW - Bessel processes

KW - Gebelein's inequality

KW - Hermite and Laguerre series expansions

KW - Itô's representation theorem

KW - Scaling property

KW - Wiener integrals

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On the construction of Wiener integrals with respect to certain pseudo-Bessel processes

Abstract

Keywords

ASJC Scopus subject areas

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