Boundary crossing problems and functional transformations for Ornstein-Uhlenbeck processes

TL;DR

This paper explores the first passage time of Ornstein-Uhlenbeck processes to time-varying thresholds, revealing connections to functional transformations and providing new explicit solutions using multiple mathematical approaches.

Contribution

It introduces a novel link between first passage times and a family of functional transformations, with three different proofs and applications to Sturm-Liouville and nonlinear differential equations.

Findings

Established connection between passage times and functional transformations.

Provided three proofs including Lie group symmetry method.

Derived explicit first passage time densities for new curve classes.

Abstract

We are interested in the law of the first passage time of an Ornstein-Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein-Uhlenbeck bridge. We provide three different proofs of this connection. The first one is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss-Markov processes and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm-Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.