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

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 by imposing a mild condition. Such results may be used for the modeling of heatwaves and related risk management challenges.

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

MSC 2010:

• 41A
• 60E
• 60G
• 60J

Acknowledgements

The authors are grateful to participants of the 6th International Conference of Mathematics in Finance, Kruger National Park, South Africa (August 2017), the London-Paris Bachelier Workshop on Mathematical Finance, University College London, U. K. (September 2017), the Fourth Young Researchers Meeting on BSDEs, Nonlinear Expectations and Mathematical Finance, Shanghai Jiaotong University, Shanghai, China (April 2018), the EMAp Research Seminar, Fundação Getulio Vargas, Rio de Janeiro (August 2018), and of the Seminar of the Department of Mathematical Sciences, University of Copenhagen (May 2019) for comments and suggestions. The authors thank Prof. Tomoko Matsui and the Institute of Statistical Mathematics in Tokyo as well as Prof. David Taylor and the African Institute for Financial Markets & Risk Management (AIFMRM), University of Cape Town, for facilitating aspects of this research through presentations and research visits. The authors are thankful for the suggestions for improvements provided by anonymous reviewers.