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

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

2011-08-10

arxiv.org/abs/1108.2132v1

Mathematics

Probability

25 pages

Scientific paper

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.

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