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The distribution of a perpetuity, with applications to risk theory and pension funding. (English) Zbl 0743.62101
Let $$\{C_ k\mid k=1,2,3,\dots\}$$ denote a stream of future cash flows, $$C_ k$$ being the (random) amount to be paid at time $$k$$. Let $$R_ k$$ denote the (random) rate of return for the period $$(k-1,k)$$. Put $$S_ 0=0$$. For $$k=1,2,3,\dots,$$ consider the accumulated value, at time $$k$$, of the first $$(k-1)$$ cash flows $$S_ k=(1+R_ k)(S_{k-1}+C_{k-1})$$. Also, consider $Z_ k=(S_ k+C_ k)/(1+R_ 1)\dots(1+R_ k),$ which is the present value of the first $$k$$ cash flows. The author studies the distributions of the stochastic processes $$\{S_ k\}$$ and $$\{Z_ k\}$$. Applications to risk theory and pension funding are also given.
Reviewer: E.Shiu (Winnipeg)

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 60G50 Sums of independent random variables; random walks
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##### References:
  Abramowitz M., Handbook of mathematical functions (1965)  Arnold L., Stochastic differential equations: Theory and Applications (1974) · Zbl 0278.60039  Billingsley P., Convergence of probability measures (1968) · Zbl 0172.21201  DOI: 10.1016/0167-6687(88)90106-0 · Zbl 0683.62059  Bourguignon F., J. Econ. Theory 9 pp 141–  Bowers N. L., Transactions of the Society of Actuaries 28 pp 177– (1976)  Bowers N. L., Transactions of the Society of Actuaries 31 pp 93– (1979)  DOI: 10.1016/0167-6687(82)90026-9 · Zbl 0526.62095  DOI: 10.2307/252033  DOI: 10.2307/1427243 · Zbl 0588.60056  Braun H., Scand. Actuarial J. pp 98– (1986)  DOI: 10.1007/BF01046992 · Zbl 0728.60012  DOI: 10.1214/aoms/1177728918 · Zbl 0053.27301  Dufresne D., Insurance and risk theory (1986) · Zbl 0606.62123  Dufresne, D. Comparison of funding methods in a static environment. Transactions of the Twenty-Third International Congress of Actuaries. Helsinki. Vol. 2, pp.99–114.  Dufresne D., Journal of the Institute of Actuaries 115 pp 535– (1988)  DOI: 10.1016/0167-6687(89)90056-5 · Zbl 0704.62096  Emmanuel D. C., Scand. Actuarial J. pp 240– (1975) · Zbl 0322.62101  Feller W., An introduction to probability theory and its applications 1, 3. ed. (1968) · Zbl 0155.23101  Feller W., An introduction to probability theory and its applications 2, 2. ed. (1971) · Zbl 0219.60003  DOI: 10.1016/0167-6687(88)90105-9 · Zbl 0657.62120  Gerber H., An introduction to mathematical risk theory (1979) · Zbl 0431.62066  Gihman I. I., Stochastic differential equations (1972) · Zbl 0242.60003  Gihman I. I., The theory of stochastic processes 3 (1979) · Zbl 0404.60061  Goldberg S., Introduction to difference equations (1986)  Grandell J., Scand. Actuarial J. pp 37– (1977) · Zbl 0384.60057  Grandell J., Scand. Actuarial J. pp 77– (1978) · Zbl 0389.62082  DOI: 10.2307/3213260 · Zbl 0364.60097  DOI: 10.1016/0304-4149(77)90051-5 · Zbl 0361.60053  Hogg R. V., Probability and statistical inference (1988)  DOI: 10.2307/3211999 · Zbl 0191.51202  Karlin S., A first course in stochastic processes (1975) · Zbl 0315.60016  Karlin S., A second course in stochastic processes (1981) · Zbl 0469.60001  Lassner F., C. R. Acad. Sc. Paris Série A 279 pp 33– (1974)  Lebedev N. N., Special functions and their applications (1972) · Zbl 0271.33001  DOI: 10.2307/3213875 · Zbl 0578.60050  Loève M., Probability theory I, 4. ed. (1977)  Loève M., Probability theory II, 4. ed. (1978)  Mandl P., Analytical treatment of one-dimensional Markov process (1968) · Zbl 0179.47802  DOI: 10.2307/2296851 · Zbl 0355.90006  DOI: 10.1016/0167-6687(86)90038-7 · Zbl 0587.62191  DOI: 10.1016/0167-6687(87)90021-7 · Zbl 0658.62124  DOI: 10.1007/978-1-4612-5254-2  Ross S. M., Stochastic processes (1983)  Ruohonen M., Scand. Actuarial J. pp 113– (1980) · Zbl 0427.62075  DOI: 10.1137/0304028 · Zbl 0143.19002  DOI: 10.1016/0304-4149(87)90017-2 · Zbl 0622.60039  DOI: 10.1007/BF02020410 · Zbl 0059.12102  DOI: 10.1007/BF02024395 · Zbl 0059.12104  Takacs L., Acta Math. Acad. Sci. Hung. 7 pp 19– (1956)  Taylor G. C., ASTIN Bulletin 10 pp 149– (1979)  Taylor J. R., Transactions of the Society of Actuaries 19 pp 1– (1967)  Treuil P., Transactions of the Society of Actuaries 33 pp 231– (1981)  Trowbridge C. L., Transactions of the Society of Actuaries 4 pp 17– (1952)  Trowbridge C. L., Transactions of the Society of Actuaries 15 pp 151– (1963) · JFM 07.0599.02  Vervaat W., Success epochs in Bernoulli trials (with applications in number of theory) (1972) · Zbl 0267.60003  DOI: 10.2307/1426858 · Zbl 0417.60073  Waters H. R., Scand. Actuarial J. pp 148– (1983) · Zbl 0517.62102  Willmot G. E., Scand. Actuarial J. pp 1– (1989) · Zbl 0679.62094  Winklevoss H. E., Pension mathematics with numerical illustrations (1977)  DOI: 10.1016/0304-4149(82)90050-3 · Zbl 0482.60062
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