# Constraint Ornstein-Uhlenbeck bridges

## TL;DR

This paper investigates the Ornstein-Uhlenbeck bridge process with a fixed area constraint, providing anticipative and non-anticipative descriptions, and extends the results to linear bridge processes.

## Contribution

It introduces both anticipative and non-anticipative models for constrained Ornstein-Uhlenbeck bridges, including a Langevin equation derivation and extension to linear bridges.

## Key findings

- Derived a stochastic differential equation for the constrained process
- Provided anticipative and non-anticipative representations
- Extended results to linear bridge processes

## Abstract

In this paper, we study the Ornstein-Uhlenbeck bridge process (i.e. the Ornstein-Uhlenbeck process conditioned to start and end at fixed points) constraints to have a fixed area under its path. We present both anticipative (in this case, we need the knowledge of the future of the path) and non-anticipative versions of the stochastic process. We obtain the anticipative description thanks to the theory of generalized Gaussian bridges while the non-anticipative representation comes from the theory of stochastic control. For this last representation, a stochastic differential equation is derived which leads to an effective Langevin equation. Finally, we extend our theoretical findings to linear bridge processes.