## Cramer-Lundberg Model

Lately, I’ve been trying to learn a bit about the insurance business. A basic building block of actuarial theory is the Cramer-Lundberg model developed by Filip Lundberg in his 1903 Uppsala thesis and later expounded upon by Harald Cramer and his academic descendants. The model makes the following assumptions and adopts the following conventions:

1. The claim sizes $(X_{k})_{k=1}^{\infty}$ are positive i.i.d. random variables with finite mean $\mu$ and possibility infinite variance $\sigma^{2}\leq\infty$.
2. The claims occur at random times with $0 < T_{1} < T_{2}<\cdots$ a.s.
3. The number of claims in the interval is denoted by $[0,t]$ is denoted by $N(t):=\sup\left\{n\geq 1:T_{n}\leq t\right\}, t\geq 0$, where $\sup\emptyset=0$.
4. The inter-arrival times $Y_{1}:=T_{1}$ and $Y_{n}:=T_{n}-T_{n-1}$ for $n\geq 2$ are i.i.d. exponentially distributed with finite mean $\mathbb{E}[X_{1}]=\lambda^{-1}$.
5. The sequences $(X_{k})_{k=1}^{\infty}$ and $(Y_{k})_{k=1}^{\infty}$ are independent of one another.

These assumptions are mathematically convenient, but their empirical validity seems, to me at least, questionable.

A consequence of (4) above is that, for any $t > 0$, $N(t)$ is Poisson distributed with parameter $\lambda t$. To see this, observe that

$\displaystyle\mathbb{P}(N(t)\geq 0)=1,\displaystyle\mathbb{P}(N(t)\geq 1)=\mathbb{P}(T_{1}\leq t)=\displaystyle\int_{0}^{t}\lambda e^{-\lambda x}dx=\displaystyle 1-e^{-\lambda t}$

Set $T_{0}$, so that $T_{n}=\sum_{j=1}^{n}T_{j}-T_{j-1}$, for $n\geq 2$. By hypothesis, $T_{1},T_{2}-T_{1},\cdots,T_{n}-T_{n-1}\sim\text{exp}(\lambda)$ and independent, so the distribution of their sum is $\text{Gamma}(n,\lambda^{-1})$. I leave the proof of this lemma as an exercise to the reader. Using this result, we see that

$\begin{array}{lcl}\displaystyle\mathbb{P}(N(t)\geq n)=\mathbb{P}(T_{n}\leq t)&=&\displaystyle\dfrac{\lambda^{n}}{\Gamma(n)}\int_{0}^{t}x^{n-1}e^{-\lambda x}dx\\[.7 em]&=&\displaystyle\dfrac{1}{\Gamma(n)}\int_{0}^{\lambda t}u^{n-1}e^{-u}dx\\[.7 em]&=&\displaystyle\left[\sum_{k=1}^{n-1}\dfrac{(\lambda t)^{k}e^{-\lambda t}}{\Gamma(k+1)}+\int_{0}^{\lambda t}e^{-u}du\right]\\&=&\displaystyle\sum_{k=1}^{n-1}\dfrac{(\lambda t)^{k}e^{-\lambda t}}{k!}+1-e^{-\lambda t}\\&=&\displaystyle\sum_{k=n}^{\infty}\dfrac{(\lambda t)^{k}e^{-\lambda t}}{k!}\end{array}$

We conclude that $N(t)$ has the distribution function of a $\text{Pois}(\lambda )$ random variable. In fact, the stochastic point process $(N(t))_{t\geq 0}$ is an example of a homogeneous Poisson process with intensity $\lambda$.

I should mention that (4) can be relaxed by dropping the restriction that the interarrival times follow an exponential distribution. The model which makes the generalization and keeps assumptions (1)-(3),(5) is called the renewal model.

The aggregate claims process $(S(t))_{t\geq 0}$ is defined by

$\displaystyle S(t)=\begin{cases}{\sum_{j=1}^{n}X_{j}} & {N(t)=n\geq 1}\\ 0 & {N(t)=n=0}\end{cases}$

We can use conditional formula to compute the aggregate claims probability distribution function for a counting process $N(t)$ satisfying the conditions of the renewal model. Since the $X_{j}$ are i.i.d., the aggregate claims distribution for $S(t)$, denoted by $G_{t}$, is given by

$\begin{array}{lcl}\displaystyle G_{t}(x)=\mathbb{P}(S(t)\leq x)&=&\displaystyle\sum_{n=0}^{\infty}\mathbb{P}(S(t)\leq x\mid N(t)=n)\mathbb{P}(N(t)=n)\\[.7 em]&=&\displaystyle\sum_{n=0}^{\infty}\mathbb{P}\left(\sum_{j=1}^{n}X_{j}\leq x\right)\mathbb{P}(N(t)=n)\\[.7 em]&=&\displaystyle\sum_{n=0}^{\infty}F^{n*}(x)\mathbb{P}(N(t)=n),\end{array}$

where $F^{n*}(x)$ is the $n$-fold convolution of $F$. If the counting process $N(t)$ is a homogeneous Poisson process with parameter $\lambda>0$, then

$\displaystyle G_{t}(x)=\sum_{n=0}^{\infty}F^{n*}(x)\dfrac{(\lambda t)^{n}e^{-\lambda t}}{n!}$