# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Abramowitz M., Handbook of mathematical functions (1965) [2] Arnold L., Stochastic differential equations: Theory and Applications (1974) · Zbl 0278.60039 [3] Billingsley P., Convergence of probability measures (1968) · Zbl 0172.21201 [4] DOI: 10.1016/0167-6687(88)90106-0 · Zbl 0683.62059 · doi:10.1016/0167-6687(88)90106-0 [5] Bourguignon F., J. Econ. Theory 9 pp 141– · doi:10.1016/0022-0531(74)90063-5 [6] Bowers N. L., Transactions of the Society of Actuaries 28 pp 177– (1976) [7] Bowers N. L., Transactions of the Society of Actuaries 31 pp 93– (1979) [8] DOI: 10.1016/0167-6687(82)90026-9 · Zbl 0526.62095 · doi:10.1016/0167-6687(82)90026-9 [9] DOI: 10.2307/252033 · doi:10.2307/252033 [10] DOI: 10.2307/1427243 · Zbl 0588.60056 · doi:10.2307/1427243 [11] Braun H., Scand. Actuarial J. pp 98– (1986) [12] DOI: 10.1007/BF01046992 · Zbl 0728.60012 · doi:10.1007/BF01046992 [13] DOI: 10.1214/aoms/1177728918 · Zbl 0053.27301 · doi:10.1214/aoms/1177728918 [14] Dufresne D., Insurance and risk theory (1986) · Zbl 0606.62123 [15] 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. [16] Dufresne D., Journal of the Institute of Actuaries 115 pp 535– (1988) · doi:10.1017/S0020268100042815 [17] DOI: 10.1016/0167-6687(89)90056-5 · Zbl 0704.62096 · doi:10.1016/0167-6687(89)90056-5 [18] Emmanuel D. C., Scand. Actuarial J. pp 240– (1975) · Zbl 0322.62101 · doi:10.1080/03461238.1975.10405104 [19] Feller W., An introduction to probability theory and its applications 1, 3. ed. (1968) · Zbl 0155.23101 [20] Feller W., An introduction to probability theory and its applications 2, 2. ed. (1971) · Zbl 0219.60003 [21] DOI: 10.1016/0167-6687(88)90105-9 · Zbl 0657.62120 · doi:10.1016/0167-6687(88)90105-9 [22] Gerber H., An introduction to mathematical risk theory (1979) · Zbl 0431.62066 [23] Gihman I. I., Stochastic differential equations (1972) · Zbl 0242.60003 · doi:10.1007/978-3-642-88264-7 [24] Gihman I. I., The theory of stochastic processes 3 (1979) · Zbl 0404.60061 · doi:10.1007/978-1-4615-8065-2 [25] Goldberg S., Introduction to difference equations (1986) [26] Grandell J., Scand. Actuarial J. pp 37– (1977) · Zbl 0384.60057 · doi:10.1080/03461238.1977.10405071 [27] Grandell J., Scand. Actuarial J. pp 77– (1978) · Zbl 0389.62082 · doi:10.1080/03461238.1978.10419478 [28] DOI: 10.2307/3213260 · Zbl 0364.60097 · doi:10.2307/3213260 [29] DOI: 10.1016/0304-4149(77)90051-5 · Zbl 0361.60053 · doi:10.1016/0304-4149(77)90051-5 [30] Hogg R. V., Probability and statistical inference (1988) [31] DOI: 10.2307/3211999 · Zbl 0191.51202 · doi:10.2307/3211999 [32] Karlin S., A first course in stochastic processes (1975) · Zbl 0315.60016 [33] Karlin S., A second course in stochastic processes (1981) · Zbl 0469.60001 [34] Lassner F., C. R. Acad. Sc. Paris Série A 279 pp 33– (1974) [35] Lebedev N. N., Special functions and their applications (1972) · Zbl 0271.33001 [36] DOI: 10.2307/3213875 · Zbl 0578.60050 · doi:10.2307/3213875 [37] Loève M., Probability theory I, 4. ed. (1977) · doi:10.1007/978-1-4684-9464-8 [38] Loève M., Probability theory II, 4. ed. (1978) · doi:10.1007/978-1-4612-6257-2 [39] Mandl P., Analytical treatment of one-dimensional Markov process (1968) · Zbl 0179.47802 [40] DOI: 10.2307/2296851 · Zbl 0355.90006 · doi:10.2307/2296851 [41] DOI: 10.1016/0167-6687(86)90038-7 · Zbl 0587.62191 · doi:10.1016/0167-6687(86)90038-7 [42] DOI: 10.1016/0167-6687(87)90021-7 · Zbl 0658.62124 · doi:10.1016/0167-6687(87)90021-7 [43] DOI: 10.1007/978-1-4612-5254-2 · doi:10.1007/978-1-4612-5254-2 [44] Ross S. M., Stochastic processes (1983) [45] Ruohonen M., Scand. Actuarial J. pp 113– (1980) · Zbl 0427.62075 · doi:10.1080/03461238.1980.10408645 [46] DOI: 10.1137/0304028 · Zbl 0143.19002 · doi:10.1137/0304028 [47] DOI: 10.1016/0304-4149(87)90017-2 · Zbl 0622.60039 · doi:10.1016/0304-4149(87)90017-2 [48] DOI: 10.1007/BF02020410 · Zbl 0059.12102 · doi:10.1007/BF02020410 [49] DOI: 10.1007/BF02024395 · Zbl 0059.12104 · doi:10.1007/BF02024395 [50] Takacs L., Acta Math. Acad. Sci. Hung. 7 pp 19– (1956) [51] Taylor G. C., ASTIN Bulletin 10 pp 149– (1979) · doi:10.1017/S0515036100006474 [52] Taylor J. R., Transactions of the Society of Actuaries 19 pp 1– (1967) [53] Treuil P., Transactions of the Society of Actuaries 33 pp 231– (1981) [54] Trowbridge C. L., Transactions of the Society of Actuaries 4 pp 17– (1952) [55] Trowbridge C. L., Transactions of the Society of Actuaries 15 pp 151– (1963) · JFM 07.0599.02 [56] Vervaat W., Success epochs in Bernoulli trials (with applications in number of theory) (1972) · Zbl 0267.60003 [57] DOI: 10.2307/1426858 · Zbl 0417.60073 · doi:10.2307/1426858 [58] Waters H. R., Scand. Actuarial J. pp 148– (1983) · Zbl 0517.62102 · doi:10.1080/03461238.1983.10408699 [59] Willmot G. E., Scand. Actuarial J. pp 1– (1989) · Zbl 0679.62094 · doi:10.1080/03461238.1989.10413851 [60] Winklevoss H. E., Pension mathematics with numerical illustrations (1977) [61] DOI: 10.1016/0304-4149(82)90050-3 · Zbl 0482.60062 · doi:10.1016/0304-4149(82)90050-3
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.