Hallo everybody,

is there any chance to compute the following expectation explicitly? (I don't think so. Am I right?)

$\displaystyle F(a):=\mathbb{E}[(1+a(e^{Z-r}-1))^{\gamma}]$

where

$\displaystyle Z$ is a normally distributed r.v. $\displaystyle \mathcal{N}(\mu,\sigma^2)$

$\displaystyle a \in [0,1]$ a constant

$\displaystyle r,\gamma > 0 $ constants

Actually, I want to maximize the expectation w.r.t $\displaystyle a$ over the interval $\displaystyle [0,1]$ ($\displaystyle a$ is non-random). I know that the maximum exists since $\displaystyle F$ is concave and the interval compact. But does anyone has an idea how to compute the maximum?

Thanks in advance