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Suppose I’m interested in modeling risk/volatility using the Cauchy distribution and I’d like to optimize some allocations using linear programming.

The Cauchy distribution is quadratic in nature but its derivative is linear.

Is this applicable and how can it be applied?

For example, the derivative tells us about the change in some variable. So if we find the derivatives of mean and variance, we might be able to account for changes in volatility (increasing/decreasing.)

A similar approach might use a Taylor series expansion. The terms should be additive and linear.

I hear that these approaches are called Moment Based Linearization.

This is quite a general question, so acceptable answers would include anything related to linearization of probability distributions using derivatives for purposes of optimizing risk decisions.

Edit: Cauchy was bad example; let's reference an arbitrary distribution such as normal.

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    $\begingroup$ The Cauchy distribution has no moments: en.wikipedia.org/wiki/Cauchy_distribution#Moments $\endgroup$
    – RobPratt
    Nov 27, 2023 at 19:26
  • $\begingroup$ Yeah, bad example- question still stands but with some arbitrary PDF that has linear moments $\endgroup$
    – jbuddy_13
    Nov 27, 2023 at 19:44
  • $\begingroup$ The derivatives of mean and variance of what random variable (not what type of probability distribution it has, but what real world thing it is modeling) with respect to what? There's been a lot of financial derivatives (e.g., options) modeling using models with volatility smiles or surfaces. Do you have in mid something which is going to be different and better than what's already out there? $\endgroup$ Nov 27, 2023 at 19:51
  • $\begingroup$ So the mean and variance, wrt $x_i$. As it's the specific $x_i$ included in the allocation that are of interest. While Cauchy was a bad choice, the standard variance definition has a linear derivative $\frac{d}{d_{x_i}}$ $\endgroup$
    – jbuddy_13
    Nov 27, 2023 at 19:55
  • $\begingroup$ I think you need to rewrite the question to be clear and explicit about everything. Th current questions and comments are not clear to me, and I doubt to anyone else. $\endgroup$ Nov 28, 2023 at 11:48

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