My question is based on the beginning of Chapter 8.3.2 in the book "Modelling Extremal Events" by Embrechts,Klüppelberg and Mikosch. We consider a Cramer-Lundberg-Model and assume that the conditions of the Cramer-Lundberg-Theorem are satiesfied.
More specifically: Let $X$ be a positive random variable with distribution function $F$ and mean $E[X] = \mu$, $Y$ a random variable, which follows an $Exp(\lambda)$-distribution independent of $X$, $c > 0$ a constant. We set $Z:=X-cY$.
We assume that the Lundberg-Exponent exists, i.e. a $\nu > 0$, that satisfies $ \int_{0}^{\infty} xe^{\nu x}(1-F(x))dx = \frac{c}{\lambda} $ Further assume that $X$ has a moment generating function, which is finite in some neighbourhood of $0$.
Now consider $\kappa(s):= E\left[e^{sZ}\right]$. My goal is to show that $\kappa(\nu) = 1$, which is stated in the book without further explaination.
I get that: $ \kappa(\nu) = E\left[e^{\nu Z}\right] = E\left[e^{\nu X}e^{-\nu cY}\right] = E\left[e^{\nu X}\right] E\left[e^{-\nu cY}\right] = E\left[e^{\nu X}\right] \frac{\lambda}{\lambda + \nu c} $ but I am stuck at this point and don't see how to use the equation above to show that this is equal to $1$.
Any help is appreciated :)