I am trying to solve a simple optimization problem related to textbook level problem in finance:
Suppose I want to find the value of $\alpha$ which minimizes $E[W(t)^{1-\gamma}]$, where
$W\left(t\right) = 1 + \alpha\left(\mu_1 - \mu_2\right)t + \alpha\sigma_1\eta\sqrt{t}$,
for $\eta\sim\mathcal{N}(0,1)$ and assuming $\mu_1 > \mu_2$ and $\gamma>1$.
Using Taylor expansion I get that for $t \ll 1$, $\alpha^* \approx \frac{\mu_1-\mu_2}{\gamma\left(\left(\mu_1-\mu_2\right)^2t + \sigma_1^2\right)}$.
However, I am interested in the sign of $d\alpha^*/dt$ (also when $t \ll 1$). Taking the derivative directly from the approximation above yields the same sign, regardless of the values of $\mu_1$, $\mu_2$, $\sigma_1$, $\gamma$ and $t$. However, I know from simulations it is not true and I can get both $d\alpha^*/dt > 0 $ and $d\alpha^*/dt < 0$.
Any ideas? No simple approximation seems to work.
Thanks