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Markov Chain Monte Carlo simulation of the distribution of some perpetuities

Part of: Markov processes Applications Stochastic analysis

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: [email protected]
∗∗ 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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References

Chamayou, J. F. and Letac, G. (1991). Explicit stationary distributions for composition of random functions and products of random matrices. J. Theoret. Prob. 4, 3–36.Google Scholar
Delbaen, F. (1993). Consols in the CIR model. Math. Finance 3, 125–134.Google Scholar
Down, D., Meyn, S. P. and Tweedie, R. L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Prob., 23, 1671–1691.CrossRefGoogle Scholar
Dufresne, D. (1990). The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J., pp. 39–79.CrossRefGoogle Scholar
Dufresne, D. (1997). On the stochastic equation cL(X)=cL[B(X+C)] and a property of gamma distributions. Bernoulli 2, 287–291.Google Scholar
Embrechts, P. and Goldie, C. M. (1994). Perpetuities and random equations. In Asymptotic Statistics. Proc. 5th Prague Symp., September 1993, ed. Mandl, P. and Husková, M. Physica, Heidelberg, pp. 73–86.Google Scholar
Geman, H. and Yor, M. (1993). Bessel processes, Asian options, and perpetuities. Math. Finance 3, 349–375.CrossRefGoogle Scholar
Gihman, I. I. and Skorohod, A. V. (1969). Introduction to the Theory of Random Processes. Saunders, Philadelphia, PA.Google Scholar
Gjessing, H. K. (1994). Integral conditions for Skorohod stochastic differential equations. Technical report No. 24, Department of Mathematics, University of Bergen.Google Scholar
Gjessing, H. K. and Paulsen, J. (1997). Present value distributions with applications to ruin theory and stochastic equations. Stoch. Proc. Appl. 71, 123–144.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.Google Scholar
Kloeden, P. E. and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Google Scholar
Liptser, R. S. and Shiryayev, A. N. (1989). Theory of Martingales. Kluwer, Dordrecht.Google Scholar
Meyn, S. P. and Tweedie, R. L. (1993a). Markov Chains and Stochastic Stability. Springer, London.CrossRefGoogle Scholar
Meyn, S. P. and Tweedie, R.L. (1993b). Stability of Markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Prob. 25, 487–517.CrossRefGoogle Scholar
Nilsen, T. and Paulsen, J. (1996). On the distribution of a randomly discounted compound Poisson process. Stoch. Proc. Appl. 61, 305–310.CrossRefGoogle Scholar
Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.CrossRefGoogle Scholar
Paulsen, J. (1993). Risk theory in a stochastic economic environment. Stoch. Proc. Appl. 46, 327–361.Google Scholar
Paulsen, J. (1997). Present value of some insurance portfolios. Scand. Actuarial J., pp. 11–37.Google Scholar
Protter, P. and Talay, D. (1997). The Euler scheme for Lévy driven stochastic differential equations. Ann. Prob. 25, 393–423.Google Scholar
Vervaat, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv Appl. Prob. 11, 750–783.CrossRefGoogle Scholar
Yor, M. (1992). Sur certaines fonctionelles exponentielles du mouvement Brownien réel. J. Appl. Prob. 29, 202–208.Google Scholar