In this section we present approximations of the risk process by
-stable Lévy motion.
We assume that claims are large, i.e.
that the distribution of their sizes is heavy-tailed.
More precisely, we let the claim sizes
distribution belong to the domain of attraction of the
-stable law with
,
see Weron (2001) and Chapter 1. This is an extension of the Brownian motion approximation approach. Note, however, that the methods and theory presented here are quite different from those used in the previous section (Weron; 1984).
We assume that claim sizes constitute an i.i.d. sequence
and that the claim counting process does not have to be independent of the sequence of the claim sizes and, in general, can be a counting renewal process constructed from the random variables having a finite second moment.
This model can be applied when claims are caused by earthquakes, floods, tornadoes, and other
natural disasters. In fact, the catastrophic losses dataset studied in Chapter 13 reveals a very heavy-tailed nature of the severity distribution. The best fit was obtained for a Burr law with
and
, which indicates a power-law decay of order
of the claim sizes distribution. Naturally, such a distribution belongs to the domain of attraction of the
-stable law with
.
We construct a sequence of risk
processes converging weakly to the -stable Lévy motion.
Let
be a sequence of risk processes defined as follows:
Let
be the
-stable Lévy motion with
a linear drift
Assumption (16.15) is satisfied for a wide class of point
processes. For example, if the times between consecutive claims
constitute i.i.d. sequence with the distribution possessing a finite
second moment. We should also notice that the skewness parameter
equals 1 for the process
if the random variables
are non-negative.
As in the Brownian motion approximation it can be shown that the
finite and infinite time ruin probabilities converge to the ruin
probabilities of the limit process. Thus it
remains to derive ruin probabilities for the process
defined in (16.18). We present asymptotic behavior for ruin probabilities in finite
and infinite time horizons and an exact formula for infinite time ruin probability.
An upper bound for finite time ruin probability will be shown.
First, we derive the asymptotic ruin probability for the finite time
horizon. Let be the ruin time (17.11) and
be the
-stable Lévy motion with
,
, and scale parameter
.
Then:
Using the asymptotic behavior of probability
when
for
, we get (Samorodnitsky and Taqqu; 1994, Prop. 1.2.15)
that
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(16.21) |
The asymptotic ruin probability in the finite time horizon is a lower
bound for the finite time ruin probability.
Let
be the
-stable Lévy motion with
and
or
and
.
Then for positive
,
, and
:
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(16.22) |
Now, we consider infinite time ruin probability for the -stable
Lévy motion. It turns out that for
it is possible to give an exact
formula for the ruin probability in the infinite time horizon. If
is the
-stable Lévy motion with
,
, and scale parameter
then for positive
,
, and
, Furrer (1998) showed that
In general, for an arbitrary we can obtain asymptotic behavior
for infinite time ruin probabilities when the initial capital tends to
infinity. Now, let
be the
-stable Lévy
motion with
,
, and scale parameter
. Then for positive
,
, and
we have (Port; 1989, Theorem 9):
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(16.24) |
For completeness it remains to consider the case , which
is quite different because the right tail of the distribution of the
-stable law with
does not behave like a power
function but like an exponential function (i.e. it is not a heavy tail).
Let
be the
-stable Lévy motion
with
,
, and scale parameter
.
Then for positive
,
, and
:
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(16.25) |
Let us assume that the sequence of claims is i.i.d. and their
distribution belongs to the domain of attraction of
the -stable law with
. Let
be the
following risk process
![]() |
(16.26) |
![]() |
(16.27) |
For , the standard deviation
. Hence, it is reasonable to put
in the general case. In this way we can compare the results for the two approximations.
Using (16.20) and (16.23)
we compute the finite and infinite time ruin
probabilities for different levels of initial capital, premium,
intensity of claims, expectation of claims and their scale parameter,
see Tables 16.2 and 16.3.
A sample path of the process
is depicted in Figure 16.2.
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. | |||
25 | 50 | 2 | 0 | . | 45896 | 0 | . | 94780 |
25 | 60 | 2 | 0 | . | 25002 | 0 | . | 90076 |
30 | 60 | 2 | 0 | . | 24440 | 0 | . | 90022 |
35 | 60 | 2 | 0 | . | 23903 | 0 | . | 89976 |
40 | 60 | 2 | 0 | . | 23389 | 0 | . | 89935 |
40 | 70 | 3 | 0 | . | 61235 | 0 | . | 96404 |
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. | |||
25 | 50 | 2 | 9 | . | 0273e-02 | 0 | . | 39735 |
25 | 60 | 2 | 3 | . | 7381e-02 | 0 | . | 23231 |
30 | 60 | 2 | 3 | . | 6168e-02 | 0 | . | 21461 |
35 | 60 | 2 | 3 | . | 5020e-02 | 0 | . | 20046 |
40 | 60 | 2 | 3 | . | 3932e-02 | 0 | . | 18880 |
40 | 70 | 3 | 1 | . | 1424e-01 | 0 | . | 44372 |
The results in the tables show the effects of the heaviness of the claim size distribution tails on the crucial parameter for insurance companies - the ruin probability.
It is clearly visible that a decrease of increases the ruin probability.
The tables also illustrate the relationship between the ruin probability and the initial capital
, premium
, intensity of claims
, expectation of claims
and their scale parameter
.
For the heavy-tailed claim distributions the ruin probability is considerably higher than for
the light-tailed claim distributions.
Thus the estimation of the stability parameter
from real data is crucial
for the choice of the premium
.