# Pollaczek–Khinchine formula for claims with expotential distribution - derivation

I am trying to understand ruin probability formula using Pollaczek–Khinchine formula described here: http://en.wikipedia.org/wiki/Ruin_theory

$$\psi(x)=\left(1-\frac{\lambda \mu}{c}\right)\sum_{n=0}^{\infty}\left(\frac{\lambda \mu}{c}\right)^{n}\left[1-F_{l}^{*n}(x)\right] (*)$$

According to wikipedia, assuming that $F$ is c.d.f. of expotential distribution, this formula simplifies to

$$\psi(x)=\frac{\lambda \mu}{c}e^{-\left(\frac{1}{\mu}-\frac{\lambda}{\beta}\right)x} (**)$$

So first I computed $F_{l}(x)$ using formula $$F_{l}(x)=\frac{1}{\mu}\int_{0}^{x}[1-F(u)]du$$

It turns out that $F_{l}(x)=F(x)=1-e^{-\frac{1}{\mu}x}$.

So the moment generation function is equal to $$F_{l}^{*}(x)=(1-\mu x)^{-1}$$ Applying formula $(*)$ we have $$\psi(x) = \left(1-\frac{\lambda \mu}{c}\right)\sum_{n=0}^{\infty}\left(\frac{\lambda \mu}{c}\right)^{n}\left[1-\frac{1}{1-\mu x}\right]^{n}=\frac{1-\frac{\lambda \mu}{c}}{1-\frac{\lambda \mu}{c}\cdot \frac{\mu x}{1-\mu x}}$$

which is different than formula (**).

What is wrong in these computation? Thanks in advance.

• What is $\beta$, why does it appear in $(**)$ but nowhere else? – Ilya Dec 3 '14 at 8:43
• I have no idea what is $\beta$, it is in a formula from wikipedia webpage mentioned above. :( – Paweł Orliński Dec 3 '14 at 8:47
• Actually, it is neither specified what are $F$ and $\mu$. Have you tried other sources regarding this formula? That is a good bunch of literature on ruins (Asmussen, Mikosch etc.) and it's very likely this formula and its exponential case are discussed there. – Ilya Dec 3 '14 at 8:50
• $F$ is c.d.f. of claim size distribution and $\mu$ is expected value of claim. – Paweł Orliński Dec 3 '14 at 17:13