Fat Tailed risks: do they get fatter when we linearize non-linear systems? I'm concerned about the risks that might arise when one decide to model a complex process into a linear or non-linear polynomial or maybe a state space model.
In the beginning of the century, Maxwell showed that the governors that were used to control motors caused sometimes a failure for the latter. These events are associated with Fat Tailed risks. I'm assuming that the same heuristic might be true in modelling complex processes (e.g. Industrial Processes such as Nuclear Reactor) into linear or non-linear polynomials.
My question: are there any papers that describe such behavior? Appreciate any help.
 A: Absolutely! There are many, such as for example:
Ahlfeld R, Belkouchi B, Montomoli F, 2016, SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos, Journal of Computational Physics, Vol:320, ISSN:0021-9991, Pages:1-16
Montomoli FF, Amirante DD, Hills NN, Shahpar SS, Massini MM. Uncertainty Quantification, Rare Events, and Mission Optimization: Stochastic Variations of Metal Temperature During a Transient. ASME. J. Eng. Gas Turbines Power. 2014;137(4):042101-042101-9. doi:10.1115/1.4028546
Ahlfeld R, Montomoli F, Scalas E, Shahpar S, 2016, Uncertainty Quantification for Fat-Tailed Probability Distributions in Aircraft Engine Simulations, Journal of Propulsion and Power 
Montomoli, F and Massini, M, Gas turbines and uncertainty quantification: Impact of PDF tails on UQ predictions, the Black Swan, GT2013-94306, ASME Turbo Expo 2013
Note that there's nothing inherently wrong in using polynomials to model Fat-Tailed risk - the problem is in the pdf of the predictors. Consider a lognormal distribution: strictly speaking this is not fat-tailed, since the tail goes to zero faster than a power, but just heavy-tailed (the tail goes to zero slower than an exponential). The log-normal distribution has an associated family of orthogonal polynomials, but as described in this answer, they are not dense in the space of mean-square integrable functions. This means that expanding the response of a complex system, whose input variables are log-normally distributed, in a series of polynomials which are orthogonal w.r.t. the log-normal measure, is not a good idea: the expansion may not converge or converge to a limit which is not actually the response function of the system. 
However, even with a fat-tailed distribution of the inputs (for example the Cauchy distribution), you can still define numerically a set of polynomials which are orthogonal to the input distribution, as long as you truncate the distribution to some limit. This is an approximation, of course, but it could make sense. Example: you want to model the possibility of a temperature spike at the inlet of a gas turbine nozzle. It might make sense to assume that rare events (very high temperature spikes) have a frequency which is much higher than expected, if the distribution of temperature spikes were Gaussian. Thus, you may want to use a lognormal, t-Student or Cauchy distribution (t-Student with 1 degree of freedom). On the other hand, you clearly don't need to model the risk posed by spikes of, say, $10^4 \ °K$: even if, from a theoretical point of view, the untruncated Cauchy distribution would give a "high" (w.r.t. a Normal distribution) probability of such spikes, in practice there's no way at all that you would reach a temperature of $10^4 \ °K$ inside a gas turbine, except pheraps if the gas turbine were hit by an atomic bomb :) Thus, you can safely truncate the Cauchy to a limit which is sufficienly higher than any physically plausible temperature, but lower than $\infty$.
A: In the paper (Uryasev, 2000) was introduced a new risk measure - Conditional Value-at-Risk (CVaR). I think you can use the CVaR as a measure of fat tail risk. 
Reference
Uryasev, S. (2000),  Conditional Value-at-Risk: Optimization Algorithms and Applications. Financial Engineering News, No. 14, February, 2000, 1-5. 
