### 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 ∫_{0} ^{1} h (s) d over(R, -)_{s}, for every h ∈ L^{2} ([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 | |

Publication status | Published - 2006 Dec |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics, Probability and Uncertainty
- Mathematics(all)
- Statistics and Probability
- Modelling and Simulation

### Cite this

*Stochastic Processes and their Applications*,

*116*(12), 1690-1711. https://doi.org/10.1016/j.spa.2006.05.001

**On the construction of Wiener integrals with respect to certain pseudo-Bessel processes.** / Funaki, Tadahisa; Hariya, Y.; Hirsch, F.; Yor, M.

Research output: Contribution to journal › Article

*Stochastic Processes and their Applications*, vol. 116, no. 12, pp. 1690-1711. https://doi.org/10.1016/j.spa.2006.05.001

}

TY - JOUR

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

AU - Funaki, Tadahisa

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 ∫0 1 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 ∫0 1 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

UR - http://www.scopus.com/inward/record.url?scp=33750630582&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33750630582&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2006.05.001

DO - 10.1016/j.spa.2006.05.001

M3 - Article

AN - SCOPUS:33750630582

VL - 116

SP - 1690

EP - 1711

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 12

ER -