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In-Depth Information
Figure
.
.
[his figure also appears in the color insert.] A Poisson-driven risk process (discontinuous
thin lines) and its Brownian motion approximation (continuous thin lines). he quantile lines enable an
easy and fast comparison of the processes. he thick solid lines representthesample
.
,...,
.
-quantile
lines based on
trajectories of the risk process, whereas the thick dashed lines correspond to their
approximation counterparts. he parameters of the risk process are the same as in Fig.
.
. From the
Ruin Probabilities Toolbox
We now return to the PCS dataset. To study the evolution of the risk process we
simulate sample trajectories and compute quantile lines. We consider a hypotheti-
cal scenario where the insurance company insures losses resulting from catastrophic
events in the United States. he company's initial capital is assumed to be u
=
billion USD and the relative safety loading used is θ
.
. We choose two differ-
ent models of the risk process based on the results from a statistical analysis (see
Sect.
.
): a homogeneous Poisson process (HPP) with log-normal claim sizes, and
a renewal process with Pareto claim sizes and log-normal waiting times. he results
arepresentedinFig.
.
.hethicksolidlineisthe“real”riskprocess,i.e.,atrajectory
constructed from the historical arrival times and values of the losses. he different
shapes of the “real” risk process in the subplots are due to the different forms of the
premium function c
=
. he thin solid line is a sample trajectory. he dotted lines
are the sample
.
,
.
,
.
,
.
,
.
,
.
,
.
,
.
,
.
-quantile lines based
on
trajectories of the risk process. he quantile lines visualize the evolution of
the density of the risk process. Clearly, if the claim severities are Pareto-distributed
then extreme events are more likely to happen than in the log-normal case, where
the historical trajectory falls outside even the
.
-quantile line. Figure
.
sug-
gests that the second model (Pareto-distributed claim sizes and log-normal waiting
times) yields a reasonable model for the “real” risk process.
(
t
)
