Q:

Risk process in Insurance

Let
$R(t) = u + ct - \sum_{k=1}^{X(t)}Z_{k}, t\geq 0 $,
be a risk process, where $u> 0$ is the initial capital of the insurance company and $c> 0$ is a premium rate. We know that the number of claims received by the insurance company is in accordance with a homogenous Poisson Process $X(t), t\geq 0$, with parameter (intensity) $\lambda > 0$. Assume that the amounts $Z_k, k = 1, 2, ...$ of successive claims are independent random variables with common probability density function $f_{Z_k}(z), z \geq 0$. Suppose that random variables $Z_{k}, k = 1, 2, ...$ are independent of $X(t)$.
I would first like to be able to compute the ranges of $q, q \geq 0$ such that the function $f(z)=q\alpha \exp (-\alpha z) + (1 - q)\beta \exp (-\beta z); \beta > \alpha > 0,$ may be taken as $f_{Z_k}(z), z \geq 0$ in the risk process.
Then, I would like to be able to compute $E[R(t)]$, and $Var[R(t)]$ in terms of the parameters $u, c, \lambda, \alpha, \beta$ and $q$.
Can anyone help? I'm pretty stuck with this problem.

A:

I'll start with some suggestions that should work.  If you're still stuck after trying them, just comment under my answer, and I'll give more detail.

For the range of $q$, it sounds like you want to find the values of $q$ that make $f(z)$ a legitimate probability density function.  If so, you want to find the range of $q$ values for which $f(z) \geq 0$ and $\int_0^{\infty} f(z) dz = 1$.  That should be doable.
For $E[R(t)]$, use the fact that $E[R(t)] = E[E[R(t)|X(t)]]$.  Knowing the distribution of $X(t)$ should make this doable as well.
For $Var[R(t)]$, try $Var[R(t)] = E[Var[R(t)|X(t)]] + Var[E[R(t)|X(t)]]$.

