A Class of Bridges of Iterated Integrals of Brownian Motion Related to Various Boundary Value Problems Involving the One-Dimensional Polyharmonic Operator. (13th December 2011) - Lachal, Aimé
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, withn being a positive integer:X n ( t ) = ∫ 0 t ( (t − s) n − 1 / ( n − 1 ) ! ) d B ( s ) for anyt ∈ [ 0, 1 ]. In this paper we construct several bridges between times0 and1 of the process( X n ( t ) ) t ∈ [ 0, 1 ] involving conditions on the successive derivatives ofX n at times0 and1 . 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.