By Bjoern Sundt, Raluca Vernic
ISBN-10: 3540928995
ISBN-13: 9783540928997
ISBN-10: 3540929002
ISBN-13: 9783540929000
Since 1980, equipment for recursive review of combination claims distributions have acquired huge consciousness within the actuarial literature.
This booklet provides a unified survey of the speculation and is meant to be self-contained to a wide quantity. because the technique is acceptable additionally outdoors the actuarial box, it truly is awarded in a basic environment, yet actuarial purposes are used for motivation.
The e-book is split into components. half I is dedicated to univariate distributions, while partially II, the technique is prolonged to multivariate settings.
Primarily meant as a monograph, this booklet is also used as textual content for classes at the graduate point. advised outlines for such classes are given.
The booklet is of curiosity for actuaries and statisticians operating in the coverage and finance undefined, in addition to for individuals in different fields like operations examine and reliability theory.
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Additional resources for Recursions for Convolutions and Compound Distributions with Insurance Applications
Sample text
4 If F is a univariate distribution, then F M∗ ≥M F. (M = 1, 2, . . 7 If p ∈ P10 and H is a univariate distribution, then μp (1) H . 4 gives ∞ p∨H = ∞ n∗ n=0 p(n)H ∞ = p(n) H n∗ ≥ n=0 p(n)n H = μp (1) H. n=0 In the following theorem, we apply the stop loss transform of the same distribution to obtain an upper and a lower bound of the stop loss transform of another distribution. 8 Let F and G be univariate distributions with finite mean, and assume that F ≤ G. 46) with ε = μF (1) − μG (1). 42) immediately gives that G ≤ On the other hand, for any real number x, we have F (x) = μF (1) − x −∞ F.
Using generating functions seems more like going from one place to another by an underground train; you get where you want, but you do not have any feeling of how the landscape gradually changes on the way. 2 based on such functions. After that, we shall deduce an alternative recursion for f based on the form of τh . 2 We have ∞ τp (s) = ∞ s n p(n) = n=0 sn n=0 λn −λ e = e−λ n! ∞ n=0 (sλ)n , n! that is, τp (s) = eλ(s−1) . 30), we obtain τf (s) = τp (τh (s)) = eλ(τh (s)−1) . 15) x=0 from which we obtain ∞ ∞ x=1 ∞ ∞ s x xf (x) = λ ∞ ys x+y h(y)f (x) = λ y=1 x=y y=1 x=0 ∞ = x yh(y)f (x − y).
As explained in Sect. 1, if this portfolio is covered by stop loss reinsurance with retention x, then the reinsurer pays (X − x)+ and the cedant pays min(X, x). Let F (x) = E(X − x)+ and F (x) = E min(X, x). We call the functions F and F the stop loss transform and retention transform of F . We define these functions for all real numbers and all distributions on the real numbers although in reinsurance applications, the distribution will normally be restricted to the non-negative numbers. For simplicity, we assume that μF (1) exists and is finite.