By Paul Embrechts
ISBN-10: 3540609318
ISBN-13: 9783540609315
"A reader's first effect on leafing via this ebook is of the massive variety of graphs and diagrams, used to demonstrate shapes of distributions...and to teach genuine info examples in a number of methods. a better studying unearths a pleasant mixture of idea and functions, with the copious graphical illustrations alluded to. one of these mix is naturally pricey to the guts of the utilized probabilist/statistician, and will galvanize even the main ardent theorists." --MATHEMATICAL stories
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Additional info for Modelling Extremal Events: for Insurance and Finance (Stochastic Modelling and Applied Probability)
Sample text
26). Can one safely interchange limits and sums? 6. 26) is the natural ruin estimate whenever F I is regularly varying. Below we shall show that a similar estimate holds true for a much wider class of dfs. 26) can be reformulated as follows. e. Z1 F (y) dy u ! 6 we obtain the following typical claim size distributions covered by the above result: { Pareto { Burr { loggamma { truncated stable distributions. 26) was the property FIn (x) nF I (x) for x ! 1 and n 2. This naturally leads us to a class of dfs which allows for a very general theory of ruin estimation for large claims.
13. 36). Then, for F 2 S , Gt (x) tF (x) x ! 1 : 46 1. 38) 0 +t +t Seal 572, 573] stresses that, apart from the homogeneous Poisson process, this process is the main realistic model for the claim number distribution in insurance applications. 38), by using Stirling's formula ; (x + 1) 2 x (x=e)x as x ! 1, that pt (n) pn n ;1 q =; ( ) n ! 36) is ful lled, so that for F 2 S , G (x) t F (x) x ! 1 : t Recall that in the homogeneous Poisson case, EN (t) = t = var(N (t)). 39) is referred to as over{dispersion of the process (N (t)) see for instance Cox and Isham 134], p.
Hence, stated in a somewhat vague way: Under the assumption of regular variation, the tail of the maximum determines the tail of the sum. 11): 1 X (1 + );n FIn (u) u 0 (u) = 1 + n=0 R where FI (x) = ;1 0x F (y) dy is the integrated tail distribution. Under the condition F I 2 R; for some 0, we might hope that the following as- ymptotic estimate holds: 1 X (u) = ;n FIn (u) (1 + ) 1 + n=0 F I (u) F I (u) ! 25) ;1 u ! 26). Can one safely interchange limits and sums? 6. 26) is the natural ruin estimate whenever F I is regularly varying.