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

Aimé Lachal

International Journal of Stochastic Analysis, 2011, vol. 2011, 1-32

Abstract:

Let ( ð µ ( ð ‘¡ ) ) ð ‘¡ ∈ [ 0 , 1 ] be the linear Brownian motion and ( ð ‘‹ ð ‘› ( ð ‘¡ ) ) ð ‘¡ ∈ [ 0 , 1 ] the ( ð ‘› − 1 ) -fold integral of Brownian motion, with ð ‘› being a positive integer: ð ‘‹ ð ‘› ∫ ( ð ‘¡ ) = ð ‘¡ 0 ( ( ð ‘¡ − ð ‘ ) ð ‘› − 1 / ( ð ‘› − 1 ) ! ) d ð µ ( ð ‘ ) for any ð ‘¡ ∈ [ 0 , 1 ] . In this paper we construct several bridges between times 0 and 1 of the process ( ð ‘‹ ð ‘› ( ð ‘¡ ) ) ð ‘¡ ∈ [ 0 , 1 ] involving conditions on the successive derivatives of ð ‘‹ ð ‘› 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.

Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:762486

DOI: 10.1155/2011/762486

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