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Multiple barrier-crossings of an Ornstein-Uhlenbeck diffusion in consecutive periods

Jiang, Y; Macrina, A; Peters, GW; (2020) Multiple barrier-crossings of an Ornstein-Uhlenbeck diffusion in consecutive periods. Stochastic Analysis and Applications 10.1080/07362994.2020.1818581. (In press).

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Abstract

We investigate the joint distribution and the multivariate survival functions for the maxima of an Ornstein-Uhlenbeck (OU) process in consecutive time-intervals. A PDE method, alongside an eigenfunction expansion is adopted, with which we first calculate the distribution and the survival functions for the maximum of a homogeneous OU-process in a single interval. By a deterministic time-change and a parameter translation, this result can be extended to an inhomogeneous OU-process. Next, we derive a general formula for the joint distribution and the survival functions for the maxima of a continuous Markov process in consecutive periods. With these results, one can obtain semi-analytical expressions for the joint distribution and the multivariate survival functions for the maxima of an OU-process, with piecewise constant parameter functions, in consecutive time periods. The joint distribution and the survival functions can be evaluated numerically by an iterated quadrature scheme, which can be implemented efficiently by matrix multiplications. Moreover, we show that the computation can be further simplified to the product of single quadratures if the filtration is enlarged. Such results may be used for the modeling of heatwaves and related risk management challenges.

Type: Article
Title: Multiple barrier-crossings of an Ornstein-Uhlenbeck diffusion in consecutive periods
DOI: 10.1080/07362994.2020.1818581
Publisher version: https://doi.org/10.1080/07362994.2020.1818581
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Ornstein-Uhlenbeck process, first-passage-time, multiple barrier-crossings and joint survival function, time-dependent barriers, Markov process, infinite series approximation and tail convergence, quadrature and Monte Carlo schemes, numerical efficiency, 41A, 60E, 60G, 60J, HITTING TIMES, DENSITY
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10109753
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