For fat tailed distributions ( i.e. those that decay slower than an exponential), large deviations not only become relatively likely, but the way in which they are realized changes dramatically: A finite fraction of the whole sample deviation is concentrated on a single variable : large deviations are not the accumulation of many small deviations, but rather they are dominated to a single large fluctuation . The regime of large deviations is separated from the regime of typical fluctuations by a phase transition where the symmetry between the points in the sample is %DIFDELCMD < {\em %%% spontaneously broken . This phenomenon has been discussed in the context of mass transport models in physics, where it takes the form of a condensation phase transition. Yet, the phenomenon is way more general. For example, in risk management of large portfolios, it suggests that one should expect losses to concentrate on a single asset: when extremely bad things happen, it is likely that there is a single factor on which bad luck concentrates. Along similar lines, one should expect that bubbles in financial markets do not gradually deflate, but rather burst abruptly and that in the most rainy day of a year, precipitation concentrate on a given spot. Analogously, when applied to biological evolution, we 're lead to infer that , if fitness changes for individual mutations have a broad distribution, those large deviations that lead to better fit species are not likely to result from the accumulation of small positive mutations. Rather they are likely to arise from large rare jumps .
Large deviations for fat tailed distributions , i.e. those that decay slower than exponential, are not only relatively likely, but they also occur in a rather peculiar way where a finite fraction of the whole sample deviation is concentrated on a single variable . The regime of large deviations is separated from the regime of typical fluctuations by a phase transition where the symmetry between the points in the sample is %DIFDELCMD < {\em %%% spontaneously broken. For stochastic processes with a fat tailed microscopic noise, this implies that while typical realizations are well described by a diffusion process with continuous sample paths, large deviation paths are typically discontinuous. For eigenvalues of random matrices with fat tailed distributed elements, a large deviation where the trace of the matrix is anomalously large concentrates on just a single eigenvalue, whereas in the thin tailed world the large deviation affects the whole distribution. These results find a natural application to finance. Since the price dynamics of financial stocks is characterized by fat tailed increments, large fluctuations of stock prices are expected to be realized by discrete jumps. Interestingly, we find that large excursions of prices are more likely realized by continuous drifts rather than by discontinuous jumps. Indeed, auto-correlations suppress the concentration of large deviations. Financial covariance matrices also exhibit an anomalously large eigenvalue, the market mode, as compared to the prediction of random matrix theory. We show that this is explained by a large deviation with excess covariance rather than by one with excess volatility .
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. This phenomenon has been discussed in the context of mass transport models in physics, where it takes the form of a condensation phase transition. Yet, the phenomenon is way more general. For example, in risk management of
spontaneously broken. For stochastic processes with a fat tailed microscopic noise, this implies that while typical realizations are well described by a diffusion process with continuous sample paths, large deviation paths are typically discontinuous. For eigenvalues of random matrices with fat tailed distributed elements, a large deviation where the trace of the matrix is anomalously large concentrates on just a single eigenvalue, whereas in the thin tailed world the large deviation affects the whole distribution. These results find a natural application to finance. Since the price dynamics of financial stocks is characterized by fat tailed increments, large fluctuations of stock prices are expected to be realized by discrete jumps. Interestingly, we find that
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portfolios, it suggests that one should expect losses to concentrate on a single asset: when extremely bad things happen, it is likely that there is a single factor on which bad luck concentrates. Along similar lines, one should expect that bubbles in financial markets do not gradually deflate, but rather burst abruptly and that in the most rainy day of a year, precipitation concentrate on a given spot. Analogously, when applied to biological evolution, we 're lead to infer that , if fitness changes for individual mutations have a broad distribution, those large deviations that lead to better fit species are not likely to result from the accumulation of small positive mutations. Rather they are likely to arise from large rare jumps
excursions of prices are more likely realized by continuous drifts rather than by discontinuous jumps. Indeed, auto-correlations suppress the concentration of large deviations. Financial covariance matrices also exhibit an anomalously large eigenvalue, the market mode, as compared to the prediction of random matrix theory. We show that this is explained by a large deviation with excess covariance rather than by one with excess volatility