A Jump Ornstein-Uhlenbeck Bridge Based on Energy-optimal Control and Its Self-exciting Extension

TL;DR

This paper introduces a new energy-optimal control-based Ornstein-Uhlenbeck bridge model driven by a spectrally-positive subordinator, with an extension incorporating self-exciting properties, and demonstrates its application to real-world streamflow regulation.

Contribution

It develops an explicit, limit-based formulation of a spectrally-positive Ornstein-Uhlenbeck bridge using energy-optimal control with a singular terminal condition, including a self-exciting extension.

Findings

Explicit expression for the Ornstein-Uhlenbeck bridge derived

Model satisfies terminal condition both analytically and numerically

Application demonstrated on real-world streamflow data

Abstract

We study a version of the Ornstein-Uhlenbeck bridge driven by a spectrally-positive subordinator. Our formulation is based on a Linear-Quadratic control subject to a singular terminal condition. The Ornstein-Uhlenbeck bridge, we develop, is written as a limit of the obtained optimally controlled processes, and is shown to admit an explicit expression. Its extension with self-excitement is also considered. The terminal condition is confirmed to be satisfied by the obtained process both analytically and numerically. The methods are also applied to a streamflow regulation problem using a real-life dataset.