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Joint law of an Ornstein–Uhlenbeck process and its supremum

Part of: Markov processes

Published online by Cambridge University Press:  16 July 2020

Christophette Blanchet-Scalliet ,
Diana Dorobantu  and
Laura Gay
Christophette Blanchet-Scalliet*
Affiliation:
University of Lyon
Diana Dorobantu*
Affiliation:
University of Lyon
Laura Gay*
Affiliation:
University of Lyon
*
**Postal Address: University Lyon 1, ISFA, LSAF (EA 2429), France
**Postal Address: University Lyon 1, ISFA, LSAF (EA 2429), France
*Postal address: CNRS UMR 5208, Ecole Centrale de Lyon, Institut Camille Jordan, France. Email address: laura.gay@ec-lyon.fr
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Abstract

Let X be an Ornstein–Uhlenbeck process driven by a Brownian motion. We propose an expression for the joint density / distribution function of the process and its running supremum. This law is expressed as an expansion involving parabolic cylinder functions. Numerically, we obtain this law faster with our expression than with a Monte Carlo method. Numerical applications illustrate the interest of this result.

Keywords

Ornstein–Uhlenbeck processjoint lawsupremumendpointFokker–Planck equation

MSC classification

Primary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 541 - 558
Copyright
© Applied Probability Trust 2020

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