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

[摘要] 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, withnbeing a positive integer:Xn(t)=∫0t((t−s)n−1/(n−1)!)dB(s)for anyt∈[0,1].In this paper we construct several bridges between times0and1of the process(Xn(t))t∈[0,1]involving conditions on the successive derivatives ofXnat times0and1. 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.

[学科分类] 应用数学