A straightforward analytical calculation of the distribution of an annuity certain with stochastic interest rate. (English) Zbl 0893.62105

Summary: Starting from the moment generating function of the annuity certain with stochastic interest rate written by means of a time discretization of the Wiener process as an \(n\)-fold integral, a straightforward evaluation of the corresponding distribution function is obtained by letting \(n\) tend to infinity. The advantage of the present method consists in the direct calculation technique of the \(n\)-fold integral, instead of using moment calculation or differential equations, and in the possible applicability of the present method to varying annuities which could be applied to IBNR results, as well as to pension fund calculations, etc.


62P05 Applications of statistics to actuarial sciences and financial mathematics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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[1] Beckman, J.; Fuelling, C., Interest and mortality randomness in some annuities, Ime, 9, 185-196, (1990) · Zbl 0711.62100
[2] Beekman, J.; Fuelling, C., Extra randomness in certain annuity models, Ime, 10, 275-287, (1991) · Zbl 0744.62142
[3] De Schepper, A.; De Vylder, F.; Goovaerts, M.; Kaas, R., Interest randomness in annuities certain, Ime, 11, 4, 271-282, (1992) · Zbl 0778.62098
[4] De Schepper, A.; Goovaerts, M.; Delbaen, F., The Laplace transform of annuities certain with exponential time distribution, Ime, 291-294, (1992) · Zbl 0784.62091
[5] De Schepper, A.; Teunen, M.; Goovaerts, M., An analytical inversion of a Laplace transform related to annuities certain, Ime, 14, 33-37, (1994) · Zbl 0796.62092
[6] Dufresne, D., The distribution of a perpetuity, with application to risk theory and pension funding, Saj, 39-79, (1990) · Zbl 0743.62101
[7] Geman, H.; Yor, M., Bessel processes, Asian options and prepetuities, Mathematical finance, 3, 4, 349-375, (1993) · Zbl 0884.90029
[8] Khandekar, D.C.; Lawande, S.V., Feynman path integrals: some exact results and applications, Physics reports, 137, 2/3, 115-229, (1986)
[9] Vanneste, M.; Goovaerts, M.; Labie, E., The distribution of annuities, Ime, 15, 37-48, (1994) · Zbl 0814.62069
[10] Yor, M., Loi de l’indice du lacet brownien, et distribution de hartman-Watson, Z. wahrscheinl. und verwante gebiete, 53, 71-95, (1980) · Zbl 0436.60057
[11] Yor, M., On some exponential functionals of Brownian motion and the problem of Asian options, ETH, Zürich, 40-56, (1991)
[12] Yor, M., On some exponential functionals of Brownian motion, Advances in applied probability, 14, 509-531, (1992) · Zbl 0765.60084
[13] Yor, M., From planar Brownian windings to Asian options, Ime, 13, 23-34, (1993) · Zbl 0792.60074
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