Expected value of ruin time

Question

I have problem with solving tasks considering ruin theory. Do you know a good literature to learn it? One of the problems is:

Let $U_n$ be a surplus process with discrete time.

$U_n=u+X_1+\ldots +X_n$,

• $u=\frac{3}{2}$ is the initial surplus
• $X_1,\ldots ,X_n$ are i.i.d. and represents difference between premiums and compensation.

$X_i$ has geometric distribution such that:

• $\mathbb{P}(X_i=-1)=1-q,$
• $\mathbb{P}(X_i=0)=(1-q)q,$
• $\mathbb{P}(X_i=1)=(1-q)q^2,$
• $\mathbb{P}(X_i=2)=(1-q)q^3, \ldots$

where $q=\frac{3}{7}$.

Let $N=\min{\{n:U_n<0\}}$ be ruin time.

What is the expected value of $N$,

$\mathbb{E}N ?$

I wanted to use $\mathbb{E}N = \sum_\limits{n=1}^{\infty}\mathbb{P}(N>n)$, but I have failed. Is there easier way to solve it?

Thanks!