International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 762486, 32 pages
http://dx.doi.org/10.1155/2011/762486
Research Article

A Class of Bridges of Iterated Integrals of Brownian Motion Related to Various Boundary Value Problems Involving the One-Dimensional Polyharmonic Operator

Aimé Lachal1,2

1Institut Camille Jordan, CNRS UMR5208, Université de Lyon, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France
2Institut National des Sciences Appliquées de Lyon, Bâtiment Léonard de Vinci, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France

Received 8 August 2011; Accepted 4 October 2011

Academic Editor: R. Liptser

Copyright © 2011 Aimé Lachal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let ( 𝐵 ( 𝑡 ) ) 𝑡 ∈ [ 0 , 1 ] be the linear Brownian motion and ( 𝑋 𝑛 ( 𝑡 ) ) 𝑡 ∈ [ 0 , 1 ] the ( 𝑛 − 1 ) -fold integral of Brownian motion, with 𝑛 being a positive integer: 𝑋 𝑛 ∫ ( 𝑡 ) = 𝑡 0 ( ( 𝑡 − 𝑠 ) 𝑛 − 1 / ( 𝑛 − 1 ) ! ) d 𝐵 ( 𝑠 ) for any 𝑡 ∈ [ 0 , 1 ] . In this paper we construct several bridges between times 0 and 1 of the process ( 𝑋 𝑛 ( 𝑡 ) ) 𝑡 ∈ [ 0 , 1 ] involving conditions on the successive derivatives of 𝑋 𝑛 at times 0 and 1 . For this family of bridges, we make a correspondence with certain boundary value problems related to the one-dimensional polyharmonic operator. We also study the classical problem of prediction. Our results involve various Hermite interpolation polynomials.