This section will be devoted to the Brownian motion approximation in risk theory and will be based on the work of Iglehart (1969). We assume that the distribution of the claim sizes belongs to the domain of attraction of the normal law. Thus, such claims attain big values with small probabilities. This assumption will cover many practical situations in which the claim size distribution possesses a finite second moment and claims constitute an i.i.d. sequence. The claims counting process does not have to be independent of the sequence of claim sizes as it is assumed in many risk models and, in general, can be a renewal process constructed from random variables having a finite first moment.
Let us consider a sequence of risk processes defined
in the following way:
To approximate the risk process by Brownian motion, we assume
,
,
,
, and
for some
where
is independent of
. Then:
Weak convergence of stochastic processes does not imply the convergence of ruin probabilities in general. Thus, to take the advantage of the Brownian motion approximations it is necessary to show that the ruin probability in finite and infinite time horizons of risk processes converges to the ruin probabilities of Brownian motion. Let us define the ruin time:
It is also possible to determine the density distribution of the
ruin time. Let be the ruin time of the process
(16.4). Then the density
of the random variable
has the following form
The Brownian model is an approximation of the risk process in the case when the distribution of claim sizes belongs to the domain of attraction of the normal law and the assumptions imposed on the risk process indicate that from the point of view of an insurance company the number of claims is large and the sizes of claims are small.
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25 | 50 | 2 | 8 | . | 0842e-02 | 8 | . | 2085e-02 |
25 | 60 | 2 | 6 | . | 7379e-03 | 6 | . | 7379e-03 |
30 | 60 | 2 | 2 | . | 4787e-03 | 2 | . | 4787e-03 |
35 | 60 | 2 | 9 | . | 1185e-04 | 9 | . | 1188e-04 |
40 | 60 | 2 | 3 | . | 3544e-04 | 3 | . | 3546e-04 |
40 | 70 | 3 | 6 | . | 5282e-02 | 6 | . | 9483e-02 |
Let us consider a risk model where the distribution of claim sizes
belongs to the domain of attraction
of the normal law and the process
counting the number of claims is a renewal counting process
constructed from i.i.d. random variables with a finite first moment.
Let be the following risk process
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(16.8) |
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(16.9) |