# Encyclopedia of Actuarial Science by Jozef Teugels, Bj?rn Sundt

By Jozef Teugels, Bj?rn Sundt

The Encyclopedia of Actuarial technology offers a well timed and entire physique of data designed to function a necessary reference for the actuarial occupation and all comparable company and fiscal actions, in addition to researchers and scholars in actuarial technological know-how and comparable parts. Drawing at the event of top overseas editors and authors from and educational learn the encyclopedia presents an authoritative exposition of either quantitative tools and functional features of actuarial technology and coverage. The cross-disciplinary nature of the paintings is mirrored not just in its insurance of key thoughts from enterprise, economics, possibility, likelihood conception and records but additionally through the inclusion of helping themes comparable to demography, genetics, operations learn and informatics.

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An extension of Panjer’s recursion, ASTIN Bulletin 32, 283–297. H. E. (1998). Loss Models: From Data to Decisions, Wiley, New York. Latouche, G. & Ramaswami, V. (1999). Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability, SIAM, Philadelphia, PA. H. (1980). The aggregate claims distribution and stop-loss reinsurance, Transactions of the Society of Actuaries 32, 523–535. H. (1981). Recursive evaluation of a family of compound distributions, ASTIN Bulletin 12, 22–26.

One of the interesting questions in insurance is the probability that the aggregate claims up to time t will excess some amount x ≥ 0. This probability is the tail probability of a compound process {S(t), t ≥ 0} and can be expressed as ∞ Pr{S(t) > x} = Pr{N (t) = n}F (n) (x), n=1 x ≥ 0, t > 0, (2) where and throughout this article, B = 1 − B denotes the tail of a distribution B. Further, expressions for the mean, variance, and Laplace transform of a compound process {S(t) = N(t) i=1 Xi , t ≥ 0} can be given in terms of the corresponding quantities of X1 and N (t) by conditioning on N (t).

Theorem 1 (Equivalent conditions for comonotonicity) A random vector X = (X1 , X2 , . . , Xn ) is comonotonic if and only if one of the following equivalent conditions holds: 1. X has a comonotonic support 2. X has a comonotonic copula, that is, for all x = (x1 , x2 , . . , xn ), we have FX (x) = min FX1 (x1 ), FX2 (x2 ), . . , FXn (xn ) ; (1) 3. For U ∼ Uniform(0,1), we have d (U ), FX−1 (U ), . . , FX−1 (U )); (2) X= (FX−1 1 2 n 4. A random variable Z and nondecreasing functions fi (i = 1, .