Multiple Barrier-Crossings of an Ornstein-Uhlenbeck Diffusion in Consecutive Periods

37 Pages Posted: 6 Mar 2019 Last revised: 18 May 2020

See all articles by Yupeng Jiang

Yupeng Jiang

University College London

Andrea Macrina

University College London; University of Cape Town (UCT)

Gareth Peters

Department of Actuarial Mathematics and Statistics, Heriot-Watt University; University College London - Department of Statistical Science; University of Oxford - Oxford-Man Institute of Quantitative Finance; London School of Economics & Political Science (LSE) - Systemic Risk Centre; University of New South Wales (UNSW) - Faculty of Science

Date Written: February 13, 2019

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 modelling 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

JEL Classification: 41A, 60E, 60G, 60J, 65C, 65D

Suggested Citation

Jiang, Yupeng and Macrina, Andrea and Peters, Gareth, Multiple Barrier-Crossings of an Ornstein-Uhlenbeck Diffusion in Consecutive Periods (February 13, 2019). Available at SSRN: https://ssrn.com/abstract=3334142 or http://dx.doi.org/10.2139/ssrn.3334142

Yupeng Jiang

University College London ( email )

Gower Street
London, WC1E 6BT
United Kingdom

Andrea Macrina

University College London ( email )

Gower Street
London, WC1E 6BT
United Kingdom

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

Gareth Peters (Contact Author)

Department of Actuarial Mathematics and Statistics, Heriot-Watt University ( email )

Edinburgh Campus
Edinburgh, EH14 4AS
United Kingdom

HOME PAGE: http://garethpeters78.wixsite.com/garethwpeters

University College London - Department of Statistical Science ( email )

1-19 Torrington Place
London, WC1 7HB
United Kingdom

University of Oxford - Oxford-Man Institute of Quantitative Finance ( email )

University of Oxford Eagle House
Walton Well Road
Oxford, OX2 6ED
United Kingdom

London School of Economics & Political Science (LSE) - Systemic Risk Centre ( email )

Houghton St
London
United Kingdom

University of New South Wales (UNSW) - Faculty of Science ( email )

Australia

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