On the concentration of large deviations for fat tailed distributions, with application to financial data

Abstract

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 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. arXiv:1201.2817v3 [cond-mat.stat-mech]

Large deviations, condensation and giant response in a statistical system

“…where K m > 1 and c is a normalization. We start with the simplest case where K m ≡ K does not depend on m [32]. In the inset of fig.1, π is plotted for K = 3/2 and different choices of N .…”

“…III. In [32] an upgraded saddle point technique based on a density functional approach was shown for a case with continuous variables related to the example (6). Another way of proceeding -still resorting to the saddle point technique -is illustrated in the Appendix.…”

“…In this case, a fluctuation N = N well above the typical value can be associated to a condensed configuration of the system [22][23][24][25][26][27][28][29][30][31][32]. This phenomenon, referred to as condensation of fluctuations, is not restricted to the particle number N but was observed for quantities as diverse as energy, exchanged heats, particles currents etc... [22][23][24][25][26][27][28][29][30][31][32]. It was shown [22,23] that in some systems condensation of fluctuations may occur because, from the mathematical point of view, asking for a specific value N = N constraints the system similarly to what a conservation law does.…”

Rare events in generalized Lévy Walks and the Big Jump principle

“…In this paper we focus on generalized Lévy walks where the event i is a jump, t i is the jump duration, T i = i−1 j=1 t j and R the particle position. In different stochastic processes, t i and R can have different interpretations (energies, masses...) [31][32][33][34].…”

Diversity waves in collapse-driven population dynamics

“…In the economics literature, the distributions of company sizes [35], as well as those of wealth of individuals [36] are known to have similar scale-free tails. Recent data for companies [35] and personal wealth [36] suggest 1/P 1.8 tail of the former distribution and 1/P 2.3 tails of the latter one. Traditionally, scale-free tails in these distributions were explained by either stochastic multiplicative processes pushed down against the lower wall (the minimal population or company size, or welfare support for low income individuals) [37][38][39], by variants of rich-get-richer dynamics [40], or in terms of Self-Organized Criticality [25,41].…”