Representations of multidimensional linear process bridges

TL;DR

This paper develops new integral and anticipative representations for multidimensional linear process bridges, including Ornstein-Uhlenbeck bridges, using transition densities and stochastic differential equations.

Contribution

It introduces a unified approach to derive representations and properties of multidimensional linear process bridges from transition densities.

Findings

Derived integral and anticipative representations for multidimensional linear process bridges.

Established a stochastic differential equation for the integral representation.

Proved conditioning properties for general multidimensional linear process bridges.

Abstract

We derive bridges from general multidimensional linear non time-homogeneous processes using only the transition densities of the original process giving their integral representations (in terms of a standard Wiener process) and so-called anticipative representations. We derive a stochastic differential equation satisfied by the integral representation and we prove a usual conditioning property for general multidimensional linear process bridges. We specialize our results for the one-dimensional case; especially, we study one-dimensional Ornstein-Uhlenbeck bridges.