Aspects of Risk Theory by Jan Grandell (auth.)

By Jan Grandell (auth.)

Risk idea, which offers with stochastic versions of an assurance enterprise, is a classical software of likelihood thought. the elemental challenge in threat concept is to enquire the smash probability of the danger enterprise. ordinarily the incidence of the claims is defined by means of a Poisson approach and the price of the claims by means of a series of random variables. This e-book is a treatise of danger concept with emphasis on versions the place the incidence of the claims is defined by means of extra basic aspect tactics than the Poisson method, similar to renewal approaches, Cox tactics and basic desk bound aspect approaches. within the Cox case the potential of threat fluctuation is explicitly taken into consideration. The presentation is predicated on sleek probabilistic equipment instead of on analytic equipment. the idea is observed with discussions on functional evaluate of smash chances and statistical estimation. Many numerical illustrations of the implications are given.

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Roo t[1 _ KO(t)] = 00 implies a top process. , if there exists to < 00 such that KO(t) < 1 for t < to and KO(t) = 1 for t ~ to, then the corresponding renewal process is a top process. o EXAMPLE 42. (Yannaros). In most applications when generalizations of the Poisson process - in the direction of renewal processes - are of interest, the first generalization which comes into mind is a renewal process N with f-distributed inter-occurrence times. In that case we have t(-y-l) 1 kO(t) = - - e- t and kO(v) (23) f(-y) - (1 + v)'Y ' where 'Y is called the form parameter and where we - for simplicity have put the scale parameter equal to 1.

The underlying point process N was defined by conditional intensities, which essentially means that its FN -compensator was specified. 2 Models allowing for risk fluctuation We shall now strictly keep ourselves to "risk fluctuations" which means that the gross risk premium c is not allowed to fluctuate. The purpose of this section is to discuss the choice of the point process describing the occurrence of the claims. The discussion will be based on the general theory of point processes, which - in contrast to the "martingale approach" to point processes - might be called the "random measure approach" to point processes.

A. " We do, however, only need stationary mixed Poisson processes. o Let N be a point process on R+ and let 5k denote the epoch of the kth point. DEFINITION 29. N is called a renewal process (with inter-occurrence time distribution J{O) if the variables 51, 52 - 51, 53 - 52, ... are independent and if 52 - 5 1,53 - 52, ... have the same distribution J{o. N is called an ordinary renewal process if 51 also has distribution J{o. N is called a stationary renewal process if J{o has finite mean 1/a and if 44 2 Generalizations of the classical risk model 8 1 has distribution K given by K(t) = a lt (1 - KD(s)) ds.

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