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An agent wishes to solve his optimisation problem: $ \mbox{max}_{\theta} \ \ \mathbb{E}U(\theta S_1 + (w - \theta) + Y)$, where $S_1$ is a random variable, $Y$ a contingent claim and $U(x) = x - \frac{1}{2}\epsilon x^2$.

My problem is - how to I 'get rid' of '$\mathbb{E}$', to get something I can work with? Thanks

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    Thanks - got it out now. I think, because of a poor stats background, I keep assuming there's something more complicated and statsy to be done. I appreciate your help.2011-10-20

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Expanding the comment by Ilya: $\mathbb{E}\,U(\theta S_1 + (w - \theta) + Y) =\mathbb{E} (\theta S_1 + (w - \theta) + Y) - \frac{\epsilon}{2} \mathbb{E} \left((\theta S_1 + (w - \theta) + Y)^2\right) $ is a quadratic polynomial in $\theta $ with negative leading coefficients. Its unique point of maximum is found by setting the derivative to $0$.