# Maximising expected utility (more of a stats question than anything)

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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what do you mean with 'get rid' of expectation? do you or your agent know the distribution of $S_1$ and $Y$? – Ilya Oct 19 '11 at 16:25
Apologies, on rereading that isn't clear at all. The question is Q6 from here: dpmms.cam.ac.uk/study/II/FinancialModels/2010-2011/sfmX1_11.pdf I'm not sure how to proceed - the only similar examples I've seen before have been able to calculate what the expectation actually is, and then proceed to maximise it, but I'm not sure how to – MartinP Oct 19 '11 at 16:27
Have you tried to substitute $U$ in this expression and open the brackets using the linearity of the expectation? The shape of $U$ gives a guess that to solve that problem you may should have to find the maximum of a parabola. – Ilya Oct 19 '11 at 18:45
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. – MartinP Oct 20 '11 at 12:37

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 $0$.

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