16.2 Brownian Motion and the Risk Model for Small Claims

This section will be devoted to the Brownian motion approximation in risk theory and will be based on the work of Iglehart (1969). We assume that the distribution of the claim sizes belongs to the domain of attraction of the normal law. Thus, such claims attain big values with small probabilities. This assumption will cover many practical situations in which the claim size distribution possesses a finite second moment and claims constitute an i.i.d. sequence. The claims counting process does not have to be independent of the sequence of claim sizes as it is assumed in many risk models and, in general, can be a renewal process constructed from random variables having a finite first moment.


16.2.1 Weak Convergence of Risk Processes to Brownian Motion

Let us consider a sequence of risk processes $ R_n(t)$ defined in the following way:

$\displaystyle R_n(t)=u_n+c_n t-\sum_{k=1}^{N(nt)}Y_k^{(n)},$ (16.1)

where $ u_n$ is the initial, $ c_n$ is the premium payed by policyholders, and the sequence $ \{Y_{k}^{(n)}: k\in \mathbb{N}\,\}$ describes the consecutive claim sizes. Assume also that $ \textrm{E}Y_k^{(n)}=\mu_n$ and $ \textrm{Var}Y_k^{(n)}=\sigma_n^2$. The point process $ N=\{N(t): t\geq 0\}$ counts claims appearing up to time $ t$ that is:

$\displaystyle N(t)=\max\left\{k: \sum_{i=1}^k T_i\leq t\right\}\,,$ (16.2)

where $ \{T_k: k\in \mathbb{N}\,\}$ is an i.i.d. sequence of nonnegative random variables describing the times between arriving claims with $ \textrm{E}T_k=1/\lambda>0$. Recall that if $ T_k$ are exponentially distributed then $ N(t)$ is a Poisson process with intensity $ \lambda $.

To approximate the risk process by Brownian motion, we assume $ n^{-1/2}u_n\rightarrow u$,
$ n^{-1/2}c_n\rightarrow c$, $ n^{1/2}\mu_n\rightarrow \mu$, $ \sigma_n^2\rightarrow\sigma^2$, and $ \textrm{E}\left(Y_k^{(n)}\right)^{2+\varepsilon}\leq M$ for some $ \varepsilon>0$ where $ M$ is independent of $ n$. Then:

$\displaystyle \frac{1}{n^{1/2}}R_n(t)\stackrel{{\mathcal{L}}}{\rightarrow} u+(c-\mu\lambda)t+\sigma\lambda^{1/2}B(t)$ (16.3)

weakly in topology $ U$ (uniform convergence on compact sets). Let us denote by $ R_B(t)$ the limit process from the above approximation, i.e.:

$\displaystyle R_B(t)=u+(c-\mu\lambda)t+\sigma\lambda^{1/2}B(t) .$ (16.4)

Property (16.3) let us approximate the risk process by $ R_B(t)$ for which it is possible to derive exact formulas for ruin probabilities in finite and infinite time horizons.


16.2.2 Ruin Probability for the Limit Process

Weak convergence of stochastic processes does not imply the convergence of ruin probabilities in general. Thus, to take the advantage of the Brownian motion approximations it is necessary to show that the ruin probability in finite and infinite time horizons of risk processes converges to the ruin probabilities of Brownian motion. Let us define the ruin time:

$\displaystyle T(R)=\inf\{t> 0 : R(t)< 0\},$ (16.5)

if the set is non-empty and $ T=\infty$ in other cases. Then $ T(R_n)\rightarrow T(R_B)
$ almost surely if $ R_n\rightarrow R_B$ almost surely as $ n\rightarrow \infty$ and $ \textrm{P}\{T(R_n)<\infty\}\rightarrow \textrm{P}\{T(R_B)<\infty\}\,.
$ Thus we need to find formulas for the ruin probabilities of the process $ {R}_B$. Let $ {R}_B$ be the Brownian motion with the linear drift defined in (16.4). Then

$\displaystyle \textrm{P}\{T(R_B)<\infty\}=\exp\left\{-2\frac{u(c-\lambda\mu)}{\sigma^2\lambda} \right\}$ (16.6)

and
$\displaystyle \textrm{P}\{T(R_B)\leq t\}$ $\displaystyle =$ $\displaystyle 1-\Phi\left\{\frac{u+(c-\lambda\mu)t} {\sigma(\lambda
t)^{1/2}}\right\}$ (16.7)
  $\displaystyle +$ $\displaystyle \exp\left\{\frac{-2u(c-\lambda\mu)}{\sigma^2\lambda}\right\}
\left[1-\Phi\left\{\frac{u-(c-\lambda\mu)t}{\sigma(\lambda t)^{1/2}}\right\}\right].$  

It is also possible to determine the density distribution of the ruin time. Let $ T(R_B)$ be the ruin time of the process (16.4). Then the density $ f_T$ of the random variable $ T(R_B)$ has the following form

$\displaystyle f_T(t)=\frac{\beta^{-1}e^{\alpha\beta}}{(2\pi)^{2/3}}t^{-3/2}
\ex...
...ft[-\frac{1}{2}\{\beta^2 t^{-1}+(\alpha\beta)^2 t\}\right],
\,\,\,\,\,\,\,t>0,
$

where $ \alpha=(c-\lambda\mu)/\sigma\lambda^{1/2}$ and $ \beta=u/\sigma\lambda^{1/2}$.

The Brownian model is an approximation of the risk process in the case when the distribution of claim sizes belongs to the domain of attraction of the normal law and the assumptions imposed on the risk process indicate that from the point of view of an insurance company the number of claims is large and the sizes of claims are small.


Table 16.1: Ruin probabilities for the Brownian motion approximation. Parameters $ \mu =20, \sigma =10,$ and $ t=10$ are fixed.
$ u$ $ {c}$ $ \lambda $ $ \Psi(t)$ $ \Psi$ . 
25 50 2 8 .0842e-02 8 .2085e-02
25 60 2 6 .7379e-03 6 .7379e-03
30 60 2 2 .4787e-03 2 .4787e-03
35 60 2 9 .1185e-04 9 .1188e-04
40 60 2 3 .3544e-04 3 .3546e-04
40 70 3 6 .5282e-02 6 .9483e-02
27544 STFdiff01.xpl


16.2.3 Examples

Let us consider a risk model where the distribution of claim sizes belongs to the domain of attraction of the normal law and the process counting the number of claims is a renewal counting process constructed from i.i.d. random variables with a finite first moment. Let $ R(t)$ be the following risk process

$\displaystyle R(t)=u+ct-\sum_{n=1}^{N(t)}Y_k\,,$ (16.8)

where $ u$ is the initial capital, $ c$ is the premium income in the unit time interval and $ \{Y_k: k\in\mathbb{N}\,\}$ are i.i.d. random variables belonging to the domain of attraction of the normal law. Moreover, $ \textrm{E}Y_k = \mu$, $ \textrm{Var}Y_k=\sigma^2$ and the intensity of arriving claims is $ \lambda $ (reciprocal of the expectation of claims inter-arrivals). Thus, we obtain:

$\displaystyle \textrm{P}\{T(R)\leq t\}\approx \textrm{P}\{T(R_B)\leq t\}$ (16.9)

and

$\displaystyle \textrm{P}\{T(R)<\infty \}\approx \textrm{P}\{T(R_B)<\infty \}\,,$ (16.10)

where

$\displaystyle R_B(t)=u+(c-\mu\lambda)t+\sigma\lambda^{1/2}B(t),
$

and $ {B(t)}$ is the standard Brownian motion. Using the formulas for ruin probabilities in finite and infinite time horizons given in (16.6) and (16.7) we compute approximate values of ruin probabilities for different levels of initial capital, premium, intensity of claims, expectation of claims and their variance, see Table 16.1. A sample path of the process $ R_B(t)$ is depicted in Figure 16.1.

Figure 16.1: A sample path of the process $ R_B$ for $ u=40$, $ c=100$, $ \mu =20$, $ \sigma =10$, and $ \lambda =3$.

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