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Boundary crossing problems and functional transformations for Ornstein–Uhlenbeck processes

Part of: Parabolic equations and systems Markov processes

Published online by Cambridge University Press:  07 October 2024

Aria Ahari ,
Larbi Alili  and
Massimiliano Tamborrino Open the ORCID record for Massimiliano Tamborrino [Opens in a new window]
Aria Ahari*
Affiliation:
University of Warwick
Larbi Alili*
Affiliation:
University of Warwick
Massimiliano Tamborrino*
Affiliation:
University of Warwick
*
*Postal address: Department of Statistics, Coventry, CV4 7AL, UK.
*Postal address: Department of Statistics, Coventry, CV4 7AL, UK.
***Email address: [email protected]
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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 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.

Keywords

First passage timesLie algebrasSturm–Liouville equationsOrnstein–UhlenbeckFokker–Planck equationOrnstein–Uhlenbeck bridgeBrownian motion

MSC classification

Primary: 60J50: Boundary theory 60J60: Diffusion processes
Secondary: 35K05: Heat equation
Type
Original Article
Information
Journal of Applied Probability , Volume 62 , Issue 1 , March 2025 , pp. 395 - 421
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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