If I have that the price of some asset is given by:
$S(t)=s \times exp{(\alpha-\lambda \sigma)t} (\sigma + 1)^{N(t)}, t \ge 0$
where $s=S(0)>0, \alpha>0, \sigma > -1$ and $\lambda > 0$ are constants, and {$N(t) : t \ge 0$} is a Poisson process with intensity $\lambda$.
I'm supposed to Find $E[e^{-\alpha t} S(t)]$. So I tried to find this expectation by calculating:
$$ \mbox{E}(e^{-\alpha t} S(t)) = \int_{0}^{\infty} e^{\alpha t}s \times exp{(\alpha-\lambda \sigma)t} (\sigma + 1)^{N(t)}\:dt $$
Now I'm stuck as I am drawing a blank how to combine any terms or how to integrate this. Did I even set this up correctly, and is this integrable? Thanks for any all help
Since $N(t)$ is Poisson, which is discrete (its pdf is $\frac{e^{-\lambda t}(\lambda t)^k}{k!}$), then you need summation and not integral:
$$\mbox{E}(e^{-\alpha t} S(t)) = \sum_{k=0}^{\infty} e^{-\alpha t}s e^{(\alpha-\lambda \sigma)t} (\sigma + 1)^{k} \frac{e^{-\lambda t}(\lambda t)^k}{k!} =\\= se^{-\lambda\sigma t}\sum_{k=0}^{\infty} \frac{e^{-\lambda t}(\lambda t(\sigma+1))^k}{k!} = \\= se^{-\lambda\sigma t}e^{\lambda t(\sigma+1)}e^{-\lambda t(\sigma+1)}\sum_{k=0}^{\infty} \frac{e^{-\lambda t(\sigma+1)}(\lambda t(\sigma+1))^k}{k!} = \\= se^{-\lambda\sigma t}e^{\lambda t}e^{-\lambda t(\sigma+1)} = \\ = se^{-2\lambda t\sigma},$$
since $$\sum_{k=0}^{\infty} \frac{e^{-\lambda t(\sigma+1)}(\lambda t(\sigma+1))^k}{k!}=1$$