Stochastic bridges of linear systems

TL;DR

This paper generalizes the concept of Brownian bridges to linear systems with inertia, modeling particles with known boundary states using conditioned Ornstein-Uhlenbeck processes and optimal control.

Contribution

It introduces a framework for stochastic bridges of linear systems, extending Brownian bridges to include velocity and higher-order dynamics with optimal control methods.

Findings

Derivation of SDEs for stochastic bridges of linear systems

Extension to higher-order linear diffusions

Application of optimal control to conditioned stochastic processes

Abstract

We study a generalization of the Brownian bridge as a stochastic process that models the position and velocity of inertial particles between the two end-points of a time interval. The particles experience random acceleration and are assumed to have known states at the boundary. Thus, the movement of the particles can be modeled as an Ornstein-Uhlenbeck process conditioned on position and velocity measurements at the two end-points. It is shown that optimal stochastic control provides a stochastic differential equation (SDE) that generates such a bridge as a degenerate diffusion process. Generalizations to higher order linear diffusions are considered.