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Perpetuities and Random Equations

  • Conference paper
Asymptotic Statistics

Part of the book series: Contributions to Statistics ((CONTRIB.STAT.))

Abstract

In life insurance and finance, stochastically discounted sums of the following type are important:

$${S_n}: = \sum\limits_{k = 1}^n {{Z_1} \cdots {Z_k}{Y_k}} .{\text{ }}(n = 1, \ldots ,\infty ) $$

where (Z i) and (Vi) are independent iid sequences satisfying mild moment conditions. Sums of this type are called perpetuities. The Y i are interpreted as payments, the Zi as discount factors. The distributional properties of S n, for n < ∞, can be studied via the following random equation:

$${S_n}\mathop = \limits^{\text{d}} \left( {{Y_1} + {S_{n - 1}}} \right){Z_1}\left( {n = 1,2, \ldots } \right),$$

where S o:= 0. In this paper we discuss some aspects of the behaviour of solutions of this and related random equations, and their applications to perpetuities.

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References

  • Aebi, M., Embrechts, P. & Mikosch, T., Stochastic discounting, aggregate claims and the bootstrap. Report, 1992, Departement Mathematik, ETH, CH-8092 Zürich, Switzerland; Adv. Appl. Probab. (March 1994) (to appear).

    Google Scholar 

  • Aldous, D., Probability Approximations via the Poisson Clumping Heuristic, Applied Math. Sciences 77, Springer-Verlag, New York, 1989. Bougerol, Ph. & Picard, N., Strict stationarity of generalized autoregressive processes, Ann. Probab. 20 (1992), 1714-1730.

    Google Scholar 

  • Brandt, A., The stochastic equation with stationary coefficients, Adv. Appl. Probab. 18 (1986), 211–220.

    Article  MATH  Google Scholar 

  • Brandt, A., Franken, P. & Lisek, B., Stationary Stochastic Models, Wiley Ser. in Probab. & Math. Stat., Wiley, Chichester, UK, 1990.

    Google Scholar 

  • Burton, R. M. & Rösler, U., An L2 convergence theorem for random affine mappings, Report, 1993, Dept. Math., Oregon State Univ., Corvallis, Oregon 97331, USA.

    Google Scholar 

  • De Schepper, A., De Vylder, F., Goovaerts, M. & Kaas, R., Interest randomness in annuities certain, Insurance: Math. and Econom. 11 (1992), 271–282.

    Article  MathSciNet  MATH  Google Scholar 

  • De Schepper, A., Goovaerts, M.& Delbaen, F., The Laplace transform of annuities certain with exponential time distributions, Insurance: Math. and Econom. 11 (1992), 291–299.

    Article  MathSciNet  MATH  Google Scholar 

  • Dufresne, D., The distribution of a perpetuity, with applications to risk theory and pension funding, Scand. Act. J. no. 1-2 (1991), 39–79.

    MathSciNet  Google Scholar 

  • Dufresne, D., On discounting when rates of return are random, in Comptes rendus du 24e Congrés International des Actuaires, Montréal, 1992.

    Google Scholar 

  • Feigin, P. D. & Tweedie, R. L., Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments, J. Time Series Anal. 6 (1985), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  • Geman, H. & Yor, M., Bessel processes, Asian options and perpetuities, Report, 1992, Dépt. Finance, ESSEC, 95021 Cergy-Pontoise, France.

    Google Scholar 

  • Goldie, C. M., Implicit renewal theory and tails of solutions of random equations, Ann. Appl. Probab. 1 (1991), 126–166.

    Article  MathSciNet  MATH  Google Scholar 

  • Grey, D. R., Regular variation in the tail behaviour of solutions of random difference equations, Report, 1993, Dept. Prob. & Stat., Univ. Sheffield, PO Box 597, Sheffield S10 2UN, UK; Ann. Appl. Probab. (to appear).

    Google Scholar 

  • Grey, D. R. & Lu Zhunwei, The fractional linear probability generating function in the random environment branching process, Report, 1992, Dept. Prob. & Stat., Univ. Sheffield, PO Box 597, Sheffield S10 2UN, UK.

    Google Scholar 

  • Grey, D. R. & Lu Zhunwei, The asymptotic behaviour of extinction probability in the Smith-Wilkinson branching process, Adv. Appl. Probab. 25 (1993), 263–289.

    Article  MATH  Google Scholar 

  • Grincevičius, A. K., On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines, Theor. Prob. Appl. 19 (1974), 163–168.

    Article  Google Scholar 

  • Grincevičius, A. K., One limit distribution for a random walk on the line, Lithuanian Math. J. 15 (1975), 580–589.

    Article  Google Scholar 

  • Grincevičius, A. K., Approxirnation-in-variation of distributions of products of random linear transformations of a straight line, Lithuanian Math. J. 18 (1978), 183–190.

    Article  Google Scholar 

  • Guegan, D., A continuous time ARCH model, Report, 1992, Inst. Galilée, Av. Jean Baptiste Clément, 93430 Vffletaneuse, France.

    Google Scholar 

  • Haan, L. de & Karandikar, R. L., Embedding a stochastic difference equation into a continuous-time process, Stoch. Proc. Appl. 32 (1989), 225–235.

    Article  MATH  Google Scholar 

  • Haan, L. de, Resnick, S. I., Rootsén, H., Vries, C. G.de, Extremal behaviour of solutions to a stochastic difference equation, with applications to ARCH processes, Stoch. Proc. Appl. 32 (1989), 213–224.

    Article  MATH  Google Scholar 

  • Kellerer, H. G., Ergodic behaviour of affine recursions, I: criteria for recurrence and transience, II: invariant measures and ergodic theorems, III: positive recurrence and null recurrence, Reports, 1992, Math. Inst. Univ. München, Theresienstrasse 39, D-8000 München, Germany. Kesten, H., Random difference equations and renewal theory for products of random matrices, Acta Math. 131 (1973), 207–248.

    Google Scholar 

  • esten, H. & Maller, R. A., Infinite limits and infinite limit points of random walks and trimmed sums, Report, 1993, Dept. Math., Univ. Western Australia, Nedlands, W.A. 6009, Australia.

    Google Scholar 

  • Letac, G., A contraction principle for certain Markov chains and its applications, in Random Matrices and their Applications (Proc. AMS-IMS-SIAM Joint Summer Research Conf. 1984) (J. E. Cohen, H. Kesten, C. E. Newman, eds.), Contemporary Mathematics 50, Amer. Math. Soc, Providence, R.I., 1986, pp. 263-273.

    Google Scholar 

  • Lindley, D. V., The theory of queues with a single server, Proc. Camb. Philos. Soc. 48 (1952), 277–289.

    Article  MathSciNet  Google Scholar 

  • Loynes, R. M., The stability of a queue with non-independent inter-arrival and service times, Proc. Camb. Philos. Soc. 58 (1962), 497–520.

    Article  MathSciNet  MATH  Google Scholar 

  • le Page, E., Théorèmes de renouvellement pour les produits de matrices aléatoires; équations aux différences aléatoires. Séminaire de Probabilités, Rennes 1983, pp. 116. Publ. Sém. Math. I, Univ. Rennes, 1983.

    Google Scholar 

  • Pakes, A. G., Some properties of a random linear difference equation, Austral. J. Statist. 25 (1983), 345–357.

    Article  MathSciNet  MATH  Google Scholar 

  • Perfekt, R., Extremal behaviour of stationary Markov chains with applications, Report, 1991, Dept. Math. Stat., Univ. Lund, S-22100 Lund, Sweden.

    Google Scholar 

  • Rachev, S. T. & Rüschendorf, L., Probability metrics and recursive algorithms, Report, 1991, Dept. Stat. & Appl. Probab., Univ. California, Santa Barbara, California 93106, USA.

    Google Scholar 

  • Rachev, S. T. & Saxnorodnitsky, G., Limit laws for a stochastic process and random recursion arising in probabilistic modelling, Report, 1992, Sunnol of O.R. & I.E., ETC Bldng., Cornell Univ., Ithaca, NY 14853-3801, USA.

    Google Scholar 

  • Rachev, S. T. & Todorovic, P., On the rate of convergence of some functionals of a stochastic process, J. Appl. Probab. 27 (1990), 805–814.

    Article  MathSciNet  MATH  Google Scholar 

  • Resnick, S. I. & Willekens, E., Moving averages with random coefficients and random coefficient autoregressive models, Commun. Stat.-Stoch. Models 7 (1991), 511–525.

    Article  MathSciNet  MATH  Google Scholar 

  • Rootsén, H., Extreme value theory for moving average processes, Ann. Probab. 14 (1986), 612–652.

    Article  MathSciNet  Google Scholar 

  • Vervaat, W., On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Adv. Appl. Probab. 11 (1979), 750–783.

    Article  MathSciNet  MATH  Google Scholar 

  • Wolfe, S. J., On a continuous analogue of the stochastic difference equation , Stoch. Proc. Appl. 12 (1982), 301–312.

    Article  MathSciNet  MATH  Google Scholar 

  • Yor, M., Some aspects of Brownian motion. Part I: Some special functionals, Lecture Notes in Mathematics, ETH-Zürich, Birkhäuser, Basel, 1992.

    Google Scholar 

  • Paul Embrechts, Department of Mathematics, ETH-Zentrum, 8092 Zürich, Switzerland Charles M. Goldie, Sunnol of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London El 4NS, UK

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, ETH-Zentrum, 8092, Zürich, Switzerland

    Paul Embrechts

  2. Sunnol of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London, El 4NS, UK

    Charles Goldie

Authors
  1. Paul Embrechts
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  2. Charles Goldie
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Editor information

Editors and Affiliations

  1. Department of Probability and Mathematical Statistics, Charles University, Sokolovská 83, CZ-186 00, Praha 8, Czech Republic

    Petr Mandl  & Marie Hušková  & 

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© 1994 Springer-Verlag Berlin Heidelberg

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Embrechts, P., Goldie, C. (1994). Perpetuities and Random Equations. In: Mandl, P., Hušková, M. (eds) Asymptotic Statistics. Contributions to Statistics. Physica, Heidelberg. https://doi.org/10.1007/978-3-642-57984-4_6

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  • DOI: https://doi.org/10.1007/978-3-642-57984-4_6

  • Publisher Name: Physica, Heidelberg

  • Print ISBN: 978-3-7908-0770-7

  • Online ISBN: 978-3-642-57984-4

  • eBook Packages: Springer Book Archive

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