A class of bridges of iterated integrals of Brownian motion related to various boundary value problems involving the one-dimensional polyharmonic operator

Abstract

Let (𝐵(𝑡))𝑡∈[0,1] be the linear Brownian motion and (𝑋𝑛(𝑡))𝑡∈[0,1] the (𝑛−1)-fold integral of Brownian motion, with 𝑛 being a positive integer: 𝑋𝑛∫(𝑡)=𝑡0((𝑡−𝑠)𝑛−1/(𝑛−1)!)d𝐵(𝑠) for any 𝑡∈[0,1]. In this paper we construct several bridges between times 0 and 1 of the process (𝑋𝑛(𝑡))𝑡∈[0,1] involving conditions on the successive derivatives of 𝑋𝑛 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.