16.3 Stable Lévy Motion and the Risk Model for Large Claims

In this section we present approximations of the risk process by $ \alpha $-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 $ \alpha $-stable law with $ 1<\alpha<2$, 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 $ \alpha= 0.4801$ and $ \tau= 2.1524$, which indicates a power-law decay of order $ \alpha\tau=1.0334$ of the claim sizes distribution. Naturally, such a distribution belongs to the domain of attraction of the $ \alpha $-stable law with $ 1<\alpha<2$.


16.3.1 Weak Convergence of Risk Processes to $ \alpha $-stable Lévy Motion

We construct a sequence of risk processes converging weakly to the $ \alpha $-stable Lévy motion. Let $ R_n(t)$ be a sequence of risk processes defined as follows:

$\displaystyle R_n(t)=u_n+c_n t-\sum_{k=1}^{N^{(n)}(t)}Y_k^{(n)}\, ,$ (16.11)

where $ u_n$ is the initial capital, $ c_n$ is the premium rate, $ \{Y_{k}^{(n)}: k\in \mathbb{N}\,\}$ is a sequence describing the sizes of the consecutive claims, and $ {N}^{(n)}(t)$, for every $ n\in \mathbb{N}$, is a point process counting the number of claims. Moreover, we assume that the random variables representing the claim sizes are of the following form

$\displaystyle Y_k^{(n)}=\frac{1}{\varphi(n)}Y_k\, ,$ (16.12)

where $ \{Y_{k}: k\in \mathbb{N}\,\}$ is a sequence of i.i.d. random variables with distribution $ F$ and expectation $ \textrm{E}Y_k = \mu$. The normalizing function $ \varphi(n)=n^{1/\alpha}L(n)$, where $ L$ is a slowly varying function at infinity. As before it is not necessary to assume that the random variables $ Y_k$ are non-negative, however, this time we assume that they belong to the domain of attraction of an $ \alpha $-stable law, that is:

$\displaystyle \frac{1}{\varphi(n)}\sum_{k=1}^{n}(Y_{k}-\mu)\stackrel{{\mathcal{L}}}{\rightarrow} Z_{\alpha,\beta}(1)\,,$ (16.13)

where $ {Z}_{\alpha,\beta}(t)$ is the $ \alpha $-stable Lévy motion with scale parameter $ \sigma'$, skewness parameter $ \beta$, and $ 1<\alpha<2$. For details see Janicki and Weron (1994) and Samorodnitsky and Taqqu (1994).

Let $ {R}_{\alpha}(t)$ be the $ \alpha $-stable Lévy motion with a linear drift

$\displaystyle R_{\alpha}(t)=u+ct-\lambda^{1/\alpha} Z_{\alpha,\beta}(t),$ (16.14)

where $ u$, $ c$, and $ \lambda $ are positive constants. Let $ \{Y_{k}\}$ be the sequence of the random variable defined above, $ \{N^{(n)}\}$ be a sequence of point processes satisfying

$\displaystyle \frac{N^{(n)}(t)-\lambda n t}{\varphi(n)}\stackrel{{\mathcal{L}}}{\rightarrow} 0,$ (16.15)

where $ \stackrel{{\mathcal{L}}}{\rightarrow}$ denotes weak convergence in the Skorokhod topology, and $ \lambda $ is a positive constant. Moreover, we assume

$\displaystyle \lim_{n\rightarrow\infty}\left(c^{(n)}-\lambda n \frac{\mu}{\varphi(n)}\right)=c$ (16.16)

and

$\displaystyle \lim_{n\rightarrow \infty}u^{(n)}=u\ .$ (16.17)

Then

$\displaystyle R_n(t)=u_n+c_n t-\frac{1}{\varphi(n)}\sum_{k=1}^{N^{(n)}(t)}Y_{k}...
...l{L}}}{\rightarrow} R_{\alpha}(t)=u+ct-\lambda^{1/\alpha}\,Z_{\alpha,\beta}(t),$ (16.18)

when $ n\to \infty$, for details see Furrer, Michna, and Weron (1997).

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 $ \beta$ equals 1 for the process $ R_{\alpha}(t)$ if the random variables $ \{Y_k\}$ are non-negative.


16.3.2 Ruin Probability in the Limit Risk Model for Large Claims

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 $ R_\alpha(t)$ 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 $ T$ be the ruin time (17.11) and $ {Z}_{\alpha,\beta}(t)$ be the $ \alpha $-stable Lévy motion with $ 0<\alpha <2$, $ -1<\beta\leq 1$, and scale parameter $ \sigma'$. Then:

$\displaystyle \lim_{u\rightarrow\infty}\frac{\textrm{P}\{T(u+cs-\lambda^{1/\alp...
...}(s))\leq t\}}{\textrm{P}\{\lambda^{1/\alpha}Z_{\alpha,\beta}(t) >u+ct\}}=1 \,,$ (16.19)

see Furrer, Michna, and Weron (1997) and Willekens (1987).

Using the asymptotic behavior of probability $ \textrm{P}\{\lambda^{1/\alpha}Z_{\alpha,\beta}(t)>u+ct\}$ when $ u\rightarrow \infty$ for $ 1<\alpha<2$, we get (Samorodnitsky and Taqqu; 1994, Prop. 1.2.15) that

$\displaystyle \textrm{P}\{T(u+cs-\lambda^{1/\alpha}Z_{\alpha,\beta}(s))\leq t\}\approx C_{\alpha}\frac{1+\beta}{2}\lambda(\sigma')^\alpha t (u+ct)^{-\alpha}\,,$ (16.20)

where

$\displaystyle C_{\alpha}=\frac{1-\alpha}{\Gamma(2-\alpha)\cos(\pi\alpha/2)}.$ (16.21)

The asymptotic ruin probability in the finite time horizon is a lower bound for the finite time ruin probability. Let $ {Z}_{\alpha,\beta}(t)$ be the $ \alpha $-stable Lévy motion with $ \alpha\neq 1$ and $ \vert\beta\vert\leq 1$ or $ \alpha =1$ and $ \beta = 0$. Then for positive $ u$, $ c$, and $ \lambda $:

$\displaystyle \textrm{P}\{T(u+cs-\lambda^{1/\alpha}Z_{\alpha,\beta}(s))\leq t\}...
...Z_{\alpha,\beta}(t)>u+ct\}} {P\{\lambda^{1/\alpha}Z_{\alpha,\beta}(t)>ct\}}\, .$ (16.22)

Now, we consider infinite time ruin probability for the $ \alpha $-stable Lévy motion. It turns out that for $ \beta =1$ it is possible to give an exact formula for the ruin probability in the infinite time horizon. If $ {Z}_{\alpha,\beta}(t)$ is the $ \alpha $-stable Lévy motion with $ 1<\alpha<2$, $ \beta =1$, and scale parameter $ \sigma'$ then for positive $ u$, $ c$, and $ \lambda $, Furrer (1998) showed that

$\displaystyle \textrm{P}\{T(u+cs-\lambda^{1/\alpha}Z_{\alpha,\beta}(s))<\infty\}= \sum_{n=0}^{\infty}\frac{(-a)^n}{\Gamma \{1+(\alpha-1)n\}} \,u^{(\alpha-1)n}\,,$ (16.23)

where $ a=c\lambda^{-1}(\sigma')^{-\alpha}\cos\{\pi(\alpha-2)/2\}$.

In general, for an arbitrary $ \beta$ we can obtain asymptotic behavior for infinite time ruin probabilities when the initial capital tends to infinity. Now, let $ {Z}_{\alpha,\beta}(t)$ be the $ \alpha $-stable Lévy motion with $ 1<\alpha<2$, $ -1<\beta\leq 1$, and scale parameter $ \sigma'$. Then for positive $ u$, $ c$, and $ \lambda $ we have (Port; 1989, Theorem 9):

$\displaystyle \textrm{P}\{T(u+cs-\lambda^{1/\alpha}Z_{\alpha,\beta}(s))<\infty\...
...^\alpha}{\alpha(\alpha-1)c}\,u^{-\alpha+1}+{\scriptstyle \cal O}(u^{-\alpha+1})$ (16.24)

when $ u\rightarrow \infty$, where

$\displaystyle A(\alpha,\beta)=\frac{\Gamma(1+\alpha)}{\pi}\sqrt{1+\beta^{2}\tan...
...rac{\pi\alpha}{2}+\arctan\{\beta\tan\left(\frac{\pi\alpha}{2}\right)\}\right].
$

For completeness it remains to consider the case $ \beta=-1$, which is quite different because the right tail of the distribution of the $ \alpha $-stable law with $ \beta=-1$ does not behave like a power function but like an exponential function (i.e. it is not a heavy tail). Let $ {Z}_{\alpha,\beta}(t)$ be the $ \alpha $-stable Lévy motion with $ 1<\alpha<2$, $ \beta=-1$, and scale parameter $ \sigma'$. Then for positive $ u$, $ c$, and $ \lambda $:

$\displaystyle \textrm{P}\{T(u+cs-\lambda^{1/\alpha}Z_{\alpha,\beta}(s))<\infty\}= \exp\{-a^{1/(\alpha-1)}u\}\, ,$ (16.25)

where $ a$ is as above.


16.3.3 Examples

Let us assume that the sequence of claims is i.i.d. and their distribution belongs to the domain of attraction of the $ \alpha $-stable law with $ 1<\alpha<2$. Let $ {R}(t)$ be the following risk process

$\displaystyle R(t)=u+ct-\sum_{n=1}^{N(t)}Y_k\,,$ (16.26)

where $ u$ is the initial capital, $ c$ is a premium rate payed by the policyholders, and $ \{Y_k: k\in\mathbb{N}\}$ is an i.i.d. sequence with the distribution belonging to the domain of attraction of the $ \alpha $-stable law with $ 1<\alpha<2$, that is fulfilling (16.13). Moreover, let $ \textrm{E}Y_k = \mu$ and the claim intensity be $ \lambda $. Similarly as in the Brownian motion approximation we obtain:

$\displaystyle \textrm{P}\{T(R)\leq t\}\approx \textrm{P}\{T(R_{\alpha})\leq t\},$ (16.27)

and

$\displaystyle \textrm{P}\{T(R)<\infty \}\approx \textrm{P}\{T(R_{\alpha})<\infty \}\,,$ (16.28)

where

$\displaystyle R_{\alpha}(t)=u+(c-\lambda\mu) t-\lambda^{1/\alpha}Z_{\alpha}(t),
$

and $ Z_{\alpha}(t)$ is the $ \alpha $-stable Lévy motion with $ \beta =1$ and scale parameter $ \sigma'$. The scale parameter can be calibrated using the asymptotic results of Mijnheer (1975), see also Samorodnitsky and Taqqu (1994, p. 50).

For $ \alpha = 2$, the standard deviation $ \sigma=\sqrt{2}\sigma'$. Hence, it is reasonable to put $ \sigma'=2^{-1/\alpha}\sigma$ 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 $ R_\alpha $ is depicted in Figure 16.2.


Table 16.2: Ruin probabilities for $ \alpha =1.0334$ and fixed $ \mu =20, \sigma =10,$ and $ t=10$.
$ u$ $ {c}$ $ \lambda $ $ \Psi(t)$ $ \Psi$ . 
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
28321 STFdiff03.xpl


Table 16.3: Ruin probabilities for $ \alpha =1.5$ and fixed $ \mu =20, \sigma =10,$ and $ t=10$.
$ u$ $ {c}$ $ \lambda $ $ \Psi(t)$ $ \Psi$ . 
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
28327 STFdiff04.xpl

Figure 16.2: A sample path of the process $ R_\alpha $ for $ \alpha =1.5$, $ u=40$, $ c=100$, $ \mu =20$, $ \sigma =10$, and $ \lambda =3$.

\includegraphics[width=1\defpicwidth]{stable.ps}

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 $ \alpha $ increases the ruin probability. The tables also illustrate the relationship between the ruin probability and the initial capital $ u$, premium $ c$, intensity of claims $ \lambda $, expectation of claims $ \mu$ and their scale parameter $ \sigma'$. 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 $ \alpha $ from real data is crucial for the choice of the premium $ c$.