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

Yupeng Jiang, Andrea Macrina, Gareth W. Peters

Research output: Contribution to journalArticle

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.
Original languageEnglish
JournalStochastic Analysis and Applications
Early online date22 Sep 2020
DOIs
Publication statusE-pub ahead of print - 22 Sep 2020

Keywords

  • math.PR

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