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**Extra randomness in certain annuity models.**
*(English)*
Zbl 0744.62142

Summary: An alternative model is presented which can be used when interest rates and future lifetimes are random for certain annuities. The Wiener stochastic process is a major component in the model, and provides much more randomness than an earlier model of the authors [ibid. 9, No. 2/3, 185-196 (1990; Zbl 0711.62100)]. Expressions for the mean values and the standard deviations of the present values of future payment streams are obtained and illustrated. Extensive boundary crossing probabilities for the Wiener stochastic process are given. Numerical comparisons with the earlier model are provided.

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

60K99 | Special processes |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

### Keywords:

random interest rates; joint randomness in interest and mortality; Wiener process; future lifetimes; annuities; mean values; standard deviations; present values of future payment streams; boundary crossing probabilities### Citations:

Zbl 0711.62100
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\textit{J. A. Beekman} and \textit{C. P. Fuelling}, Insur. Math. Econ. 10, No. 4, 275--287 (1992; Zbl 0744.62142)

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### References:

[1] | Beekman, J.A.; Fuelling, C.P., Interest and mortality randomness in some annuities, Insurance: mathematics and economics, 9, 185-196, (1990) · Zbl 0711.62100 |

[2] | Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J., Actuarial mathematics, (1986), Society of Actuaries Schaumburg, IL |

[3] | De Pril, N., The distribution of actuarial functions, Bulletin of the swiss assoc. of actuaries, 173-183, (1989) |

[4] | Dhaene, Jan, Stochastic interest rates and autoregressive integrated moving average processes, ASTIN bulletin, 19, 131-138, (1989) |

[5] | Dhaene, Jan, Distributions in life insurance, ASTIN bulletin, 20, 81-92, (1990) |

[6] | Dhaene, Jan, Actuarial functions and random rates of return, (1990), Submitted for publication |

[7] | Dufresne, F.; Gerber, H.U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: mathematics and economics, 10, 51-59, (1991) · Zbl 0723.62065 |

[8] | Feller, William, () |

[9] | Gerber, H.U., An extension of the renewal equation and its application in the collective theory of risk, Scandinavian actuarial journal, 205-210, (1970) · Zbl 0229.60062 |

[10] | Hogg, R.V.; Craig, A.T., () |

[11] | Malmquist, S., On certain confidence contours for distribution functions, Annals mathematical statistics, 25, 523-533, (1954) · Zbl 0056.37802 |

[12] | Panjer, H.H.; Bellhouse, D.R., Stochastic modelling of interest rates with application to life contingencies, Journal of risk and insurance, 47, 91-110, (1980) |

[13] | Park, C.; Schuurmann, F.J., Evaluations of barrier crossing probabilities of Wiener paths, Journal of applied probability, 13, 267-275, (1976) · Zbl 0344.60047 |

[14] | Pollard, J.H., Fluctuating interest rates revisited. pensions and insurances in a fluctuating environment, Trans. of the 23rd international congress of actuaries, 1, 439-451, (1988) |

[15] | Shepp, L.A., Radonâ€”nikodym derivatives of Gaussian measures, Annals mathematical statistics, 37, 321-354, (1966) · Zbl 0142.13901 |

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