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

الملخص EN

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)=∫0t((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.