17.2 Fractional Brownian Motion and the Risk Model of Good and Bad Periods

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 $ B_H$ is called fractional Brownian motion if for some $ 0<H\leq 1$:

$ 1.$
$ B_H(0)=0$ almost surely.
$ 2.$
$ B_H$ has strictly stationary increments, that is the random function $ M_h(t)=B_H(t+h)-B_H(t)$, $ h\geq 0$, is strictly stationary.
$ 3.$
$ B_H$ is self-similar of order $ H$ denoted $ H-ss$, that is $ {\cal{L}}\{B_H(ct)\}= {\cal{L}}\{c^H B_H(t)\}$ in the sense of finite-dimensional distributions.
$ 4.$
Finite dimensional distributions of $ B_H$ are Gaussian with $ \textrm{E}B_H(t)=0$
$ 5.$
$ B_H$ is almost surely continuous.
If not stated otherwise explicitly, we let the parameter of self-similarity satisfy $ \frac{1}{2}<H<1$. The concept of semi-stability was introduced by Lamperti (1962) and recently discussed in Embrechts and Maejima (2002). Mandelbrot and Van Ness (1968) call it self-similarity when appearing in conjunction with stationary increments, as it does here.

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 $ \{T_n^G,
n\in \mathbb{N}\}$ be i.i.d. non-negative random variables representing good periods; similarly, let $ \{S^B,
S_n^B, n\in \mathbb{N}\}$ be i.i.d. non-negative random variables representing bad periods. The $ T$'s are assumed independent of the $ S$'s, the common distribution of good periods is $ F^G$, and the distribution of bad periods is $ F^B$. We assume that both $ F^G$ and $ F^B$ have finite means $ \nu_G$ and $ \nu_B$, respectively, and we set $ \nu=\nu_G+\nu_B$.

Consider the pure renewal sequence initiated by a good period $ \{0,
\sum_{i=1}^{n}(T_i^G+S_i^B), n\in \mathbb{N}\}$. The inter-arrival distribution is $ F^G*F^B$ and the mean inter-arrival time is $ \nu$. This pure renewal process has a stationary version $ \{D, D+\sum_{i=1}^{n}(T_i^G+S_i^B), n\in \mathbb{N}\}$, where $ D$ is a delay random variable (Asmussen; 1987). However, by defining the initial delay interval of length $ D$ 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 $ D$ into a good and bad period. Define three independent random variables $ B$, $ T_0^G$, and $ S_0^B$, which are independent of $ (S^B, T_n^G, S_n^B, n\in \mathbb{N})$, as follows: $ B$ is a Bernoulli random variable with values in $ \{0,1\}$ and mass function

$\displaystyle \textrm{P}(B=1)=\frac{\nu_G}{\nu}=1-\textrm{P}(B=0)
$

and

$\displaystyle \textrm{P}(T_0^G>x)=\int_x^\infty\frac{1-F^G(s)}{\nu_G}~ds\stackrel{\mathrm{def}}{=}1-F_0^G(x),
$

$\displaystyle \textrm{P}(S_0^B>x)=\int_x^\infty\frac{1-F^B(s)}{\nu_B}~ds\stackrel{\mathrm{def}}{=}1-F_0^B(x),
$

for $ x>0$. Define a delay random variable $ D_0$ by

$\displaystyle D_0=(T_0^G+S^B)B+(1-B)S_0^B
$

and a delayed renewal sequence by

$\displaystyle \{S_n, n\geq 0\}\stackrel{\mathrm{def}}{=}\left\{D_0, D_0+\sum_{i=1}^{n}(T_i^G+S_i^B), n\geq 0\right\}.
$

One can verify that this delayed renewal sequence is stationary (Heath, Resnick, and Samorodnitsky; 1998).

We now define $ L(t)$ to be 1 if $ t$ falls in a good period, and $ L(t)=0$ if $ t$ is in a bad period. More precisely, the process $ \{L(t), t\geq 0\}$ is defined in terms of $ \{S_n, n\geq 0\}$ as follows

$\displaystyle L(t)=B I(0\leq t<T_0^G)+\sum_{n=0}^\infty I(S_n\leq t<S_n+T_{n+1}^G).$ (17.1)

The process $ \{L(t), t\geq 0\}$ is strictly stationary and

$\displaystyle \textrm{P}\{L(t)=1\}=\textrm{E}L(t)=\frac{\nu_G}{\nu}.
$

Let $ \{Y_n^G, n\geq 0\}$ be i.i.d. random variables representing claims appearing in good periods (e.g. $ Y_n^G$ describes a claim which may appear at the $ n$-th moment in a good period). Similarly, let $ \{Y_n^B, n\geq 0\}$ be i.i.d. random variables representing claims appearing in bad periods (e.g. $ Y_n^B$ describes a claim which may appear at the $ n$-th moment in a bad period). We assume that $ \{Y_n^G, n\geq 0\}$, $ \{Y_n^B, n\geq 0\}$ and $ \{L(t), t\geq 0\}$ are independent, $ E(Y_0^G)=g < E(Y_0^B)=b$, and the second moments of $ Y_0^G$ and $ Y_0^B$ exist. Then the claim $ Y_n$ appearing at the $ n$-th moment is

$\displaystyle Y_n=L(n)Y_n^G+\{1-L(n)\}Y_n^B,\ \ n\geq 0.$ (17.2)

Furthermore, the sequence $ \{Y_n, n\geq 0\}$ is stationary.

Assume that

$\displaystyle 1-F^G(t)=t^{-(C+1)}K(t),$ (17.3)

for $ t\rightarrow\infty$, $ 0<C<1$, where $ K$ is slowly varying at infinity. Moreover, assume that

$\displaystyle 1-F^B(t)={\scriptstyle \cal O}\{1-F^G(t)\},$ (17.4)

as $ t\rightarrow\infty$ and there exists an $ n\geq 1$ such that $ (F_G*F_B)^{*n}$ is nonsingular. Then

$\displaystyle \textrm{Cov}(Y_0, Y_n)\sim\frac{\nu_B^2(b-g)^2}{C\nu^3}n^{-C}K(n)$ (17.5)

when $ n\rightarrow \infty$ (Heath, Resnick, and Samorodnitsky; 1998).

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 $ \textrm{E}Y_n=\mu$ and $ \varphi(n)=n^H K(n)$, where $ K$ is a slowly varying function at infinity. Let the sequence $ \{Y_{k} :
k\in \mathbb{N}\}$ be as above and let $ \{N^{(n)} : n\in \mathbb{N}\}$ be a sequence of point processes such that

$\displaystyle \frac{N^{(n)}(t)-\lambda n t}{\varphi(n)}\stackrel{\cal{L}}{\rightarrow} 0$ (17.6)

weakly in the Skorokhod topology (Jacod and Shiryaev; 1987) for some positive constant $ \lambda $. Assume also that

$\displaystyle \lim_{n\rightarrow\infty}\left\{c^{(n)}-\lambda n \frac{\mu}{\varphi(n)}\right\}=c$ (17.7)

and

$\displaystyle \lim_{n\rightarrow \infty}u^{(n)}=u\ .$ (17.8)

Then

$\displaystyle u^{(n)}+c^{(n)}t-\frac{1}{\varphi(n)}\sum_{k=1}^{N^{(n)}(t)}Y_{k}\stackrel{{\cal{L}}}{\rightarrow} u+ct-\lambda^{H}\,B_{H}(t)$ (17.9)

in the Skorokhod topology as $ n\to \infty$. Condition (17.6) is satisfied for a wide class of point processes. For example, if the times between consecutive claims constitute an i.i.d. sequence with the distribution possessing a finite second moment.