In this section we describe fractional Brownian motion approximation in risk theory. We show that under suitable assumptions the risk process constructed from claims appearing in good and bad periods can be approximated by the fractional Brownian motion with a linear drift. Hence, we first introduce the definition of fractional Brownian motion and then construct the model.
A process is called fractional Brownian motion if for
some
:
When we observe arriving claims we assume that we have good and bad periods
(e.g. periods of good weather and periods
of bad weather). These two periods alternate. Let
be i.i.d. non-negative
random variables representing good periods; similarly, let
be i.i.d. non-negative random variables representing bad periods. The
's
are assumed independent of the
's, the common distribution of
good periods is
, and the distribution of bad periods is
. We assume that both
and
have finite means
and
, respectively, and we set
.
Consider the pure renewal sequence initiated by a good period
. The inter-arrival
distribution is
and the mean inter-arrival time is
.
This pure renewal process has a stationary version
, where
is a delay random variable (Asmussen; 1987). However, by defining the initial delay
interval of length
this way, the interval does not decompose
into a good and a bad period the way subsequent inter-arrival
intervals do. Consequently, we turn to an alternative construction
of the stationary renewal process
to decompose the delay random variable
into a good and bad period.
Define three independent random variables
,
, and
, which are independent of
,
as follows:
is a Bernoulli random variable with values in
and mass function
We now define to be 1 if
falls in a good period, and
if
is in a bad period. More precisely, the process
is defined in terms of
as
follows
Let
be i.i.d. random variables representing claims appearing in good periods
(e.g.
describes a claim which may appear at the
-th
moment in a good period). Similarly, let
be
i.i.d. random variables representing
claims appearing in bad periods (e.g.
describes a claim
which may appear at the
-th moment in a bad period). We assume
that
,
and
are independent,
, and the
second moments of
and
exist. Then the claim
appearing at the
-th moment is
Assume that
We assumed that the good period dominates the bad period but one can approach the problem reversely (i.e. the bad period can dominate the good period) because of the symmetry of the good and bad period characteristics in the covariance function.
Assume that
and
, where
is
a slowly varying function at infinity. Let the sequence
be as above and let
be a
sequence of point processes such that