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

TL;DR

This paper constructs bridges of iterated Brownian integrals with boundary conditions, linking them to boundary value problems of the polyharmonic operator and exploring related prediction problems using Hermite interpolation.

Contribution

It introduces new bridges of iterated Brownian integrals with boundary conditions, connecting stochastic processes to boundary value problems of the polyharmonic operator.

Findings

Constructed several bridges of iterated Brownian integrals with boundary conditions.

Established correspondence between these bridges and boundary value problems of the polyharmonic operator.

Analyzed prediction problems using Hermite interpolation polynomials.

Abstract

Let $(B(t))_{t\in [0,1]}$ be the linear Brownian motion and $(X_n(t))_{t\in [0,1]}$ be the $(n-1)$-fold integral of Brownian motion, $n$ being a positive integer: $$ X_n(t)=\int_0^t \frac{(t-s)^{n-1}}{(n-1)!} \,\dd B(s) for any $t\in[0,1]$. $$ In this paper we construct several bridges between times 0 and 1 of the process $(X_n(t))_{t\in [0,1]}$ involving conditions on the successive derivatives of $X_n$ at times 0 and 1. For this family of bridges, we make a correspondance 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.