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Markov Chain Monte Carlo simulation of the distribution of some perpetuities

Part of: Stochastic analysis Markov processes Applications

Published online by Cambridge University Press:  01 July 2016

Jostein Paulsen  and
Arne Hove
Jostein Paulsen*
Affiliation:
University of Bergen
Arne Hove*
Affiliation:
Risk Management, DNB, Norway
*
∗ Postal address: Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway. Email address: jostein@mi.uib.no
∗∗ Postal address: Risk Management, DNB, PO BOX 1171 Sentrum, N-0107, Oslo, Normay.
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Abstract

We study the present value Z∞ = ∫0∞ e-Xt-dYt where (X,Y) is an integrable Lévy process. This random variable appears in various applications, and several examples are known where the distribution of Z∞ is calculated explicitly. Here sufficient conditions for Z∞ to exist are given, and the possibility of finding the distribution of Z∞ by Markov chain Monte Carlo simulation is investigated in detail. Then the same ideas are applied to the present value Z-∞ = ∫0∞ exp{-∫0tRsds}dYt where Y is an integrable Lévy process and R is an ergodic strong Markov process. Numerical examples are given in both cases to show the efficiency of the Monte Carlo methods.

Keywords

Present valueLévy processMarkov chain Monte Carlo simulation

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces 62P05: Applications to actuarial sciences and financial mathematics
Secondary: 60H05: Stochastic integrals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 1 , March 1999 , pp. 112 - 134
Copyright
Copyright © Applied Probability Trust 1999 

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