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

Let be the linear Brownian motion and the -fold integral of Brownian motion, with being a positive integer: for any In this paper we construct several bridges between times and of the process involving conditions on the successive derivatives of at times and . 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. 1. Introduction Throughout the paper, we will denote, for any enough differentiable function , its th derivative by or . Let be the linear Brownian motion started at 0 and be the linear Brownian bridge within the time interval : . These processes are Gaussian processes with covariance functions For a given continuous function , the functions and , respectively, defined on by are the solutions of the respective boundary value problems on : Observe that the differential equations are the same in both cases. Only the boundary conditions differ. They are Dirichlet-type boundary conditions for Brownian bridge while they are Dirichlet/Neumann-type boundary conditions for Brownian motion. These well-known connections can be extended to the polyharmonic operator , where is a positive integer. This latter is associated with the -fold integral of Brownian motion : (Notice that all of the derivatives at time 0 naturally vanish: .) Indeed, the following facts, for instance, are known (see, e.g., [1, 2]).(i)The covariance function of the process coincides with the Green function of the boundary value problem (ii)The covariance fonction of the bridge coincides with the Green function of the boundary value problem (iii)The covariance fonction of the bridge coincides with the Green function of the boundary value problem Observe that the differential equations and the boundary conditions at 0 are the same in all cases. Only the boundary conditions at 1 differ. Other boundary value problems can be found in [3, 4]. We refer the reader to [5] for a pioneering work dealing with the connections between general Gaussian processes and Green functions; see also [6]. We also refer to [7–13] and the references therein for various properties, namely, asymptotical study, of the iterated integrals of Brownian motion and more general Gaussian processes as well as to [3, 4, 14, 15] for interesting applications of these processes to statistics. The aim of this work is to examine all the possible conditioned processes of involving different events at time 1: for a certain