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Marshall–Olkin distributions, subordinators, efficient simulation, and applications to credit risk

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  26 June 2017

Yunpeng Sun ,
Rafael Mendoza-Arriaga  and
Vadim Linetsky
Yunpeng Sun*
Affiliation:
Northwestern University
Rafael Mendoza-Arriaga*
Affiliation:
The University of Texas at Austin
Vadim Linetsky*
Affiliation:
Northwestern University
*
* Postal address: Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA.
*** Postal address: Department of Information, Risk, and Operations Management, McCombs School of Business, The University of Texas at Austin, CBA 5.202, B6500, 1 University Station, Austin, TX 78712, USA. Email address: rafael.mendoza-arriaga@mccombs.utexas.edu
* Postal address: Department of Industrial Engineering and Management Sciences, McCormick School of Engineering and Applied Sciences, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA.
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Abstract

In the paper we present a novel construction of Marshall–Olkin (MO) multivariate exponential distributions of failure times as distributions of the first-passage times of the coordinates of multidimensional Lévy subordinator processes above independent unit-mean exponential random variables. A time-inhomogeneous version is also given that replaces Lévy subordinators with additive subordinators. An attractive feature of MO distributions for applications, such as to portfolio credit risk, is its singular component that yields positive probabilities of simultaneous defaults of multiple obligors, capturing the default clustering phenomenon. The drawback of the original MO fatal shock construction of MO distributions is that it requires one to simulate 2n-1 independent exponential random variables. In practice, the dimensionality is typically on the order of hundreds or thousands of obligors in a large credit portfolio, rendering the MO fatal shock construction infeasible to simulate. The subordinator construction reduces the problem of simulating a rich subclass of MO distributions to simulating an n-dimensional subordinator. When one works with the class of subordinators constructed from independent one-dimensional subordinators with known transition distributions, such as gamma and inverse Gaussian, or their Sato versions in the additive case, the simulation effort is linear in n. To illustrate, we present a simulation of 100,000 samples of a credit portfolio with 1,000 obligors that takes less than 18 seconds on a PC.

Keywords

Marshall–Olkin multivariate exponential distributionLévy processadditive subordinatorsubordinatorsimulationdependent lifetimefailuredefaultreliabilitycredit risk

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 60J28: Applications of continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 481 - 514
Copyright
Copyright © Applied Probability Trust 2017 

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