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Consider the two following optimization problem

1) $$ \min_{\theta} \ln E_{\theta}[ e^{X}]$$

2) $$ \min_{\theta} E_{\theta}[ X]$$ with the constraint $$ Var_{\theta}[X] <c$$

Is it true that the first one is a better measure for risk-sensitivity cost as it takes other moments (which is visible from the Taylor series expansion) also into account ?

In general, can we say anything about the usefulness of the first cost compared to any other risk-sensitive cost ?

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    $\begingroup$ What is $\theta$? A probability measure? If so, from which set of measures? $Var$ is variance or value-at-risk? Where do you take these definitions from? $\endgroup$ – Ric Nov 12 '14 at 9:39
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I wouldn't say 1) is better because it's very rigid. If you want to include some higher moments (most likely you won't need more than 4th order), it's better to do it explicitly rather than to stuff the moments of all orders into the criterion.

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