Maximising exponent of a Gaussian random variable I am trying to solve the maximisation problem defined by:
$$\max_x \mathbb{E}_t \Big[ -e^{-\gamma (\omega + x (Y_{t+1} - Y_t))} \Big]   $$
where I am told that $Y_{t+1} - Y_t \sim N(0,\sigma_Y ).$ It should be straightforward to verify that
$$ x^* = \frac{\mathbb{E}(Y_{t+1} - Y_t)}{\gamma \sigma_Y}. $$
I cannot seem to back out this simple expression though, I must be making a simple algebraic mistake somewhere. I am trying to use the fact that for a lognormal we have $\mathbb{E}_t\Big[e^{(Y_{t+1}-Y_t)}\Big]=e^{\sigma_Y^2/s}.$ This hasn't helped though.

If we just get rid of the clutter, your question boils down to
$$\min_x \mathbb{E}_t \Big[ e^{-\gamma x Z} \Big] = \min_x e^{\frac{1}{2}\gamma^2 x^2 \sigma_Y^2}$$
The minimum is attained for $x^*=0$.
