  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.
