Ubiquitous Power Law Scaling in Nonlinear Self-Excited Hawkes Processes

The origin(s) of the ubiquity of probability distribution functions with power law tails is still a matter of fascination and investigation in many scientific fields from linguistic, social, economic, computer sciences to essentially all natural sciences. In parallel, self-excited dynamics is a prevalent characteristic of many systems, from the physics of shot noise and intermittent processes, to seismicity, financial and social systems. Motivated by activation processes of the Arrhenius form, we bring the two threads together by introducing a general class of nonlinear self-excited point processes with fast-accelerating intensities as a function of "tension." Solving the corresponding master equations, we find that a wide class of such nonlinear Hawkes processes have the probability distribution functions of their intensities described by a power law on the condition that (i) the intensity is a fast-accelerating function of tension, (ii) the distribution of marks is two sided with nonpositive mean, and (iii) it has fast-decaying tails. In particular, Zipf's scaling is obtained in the limit where the average mark is vanishing. This unearths a novel mechanism for power laws including Zipf's law, providing a new understanding of their ubiquity.

Electrical studies of Barkhausen switching noise in ferroelectric lead zirconate titanate (PZT) and BaTiO3: critical exponents and temperature-dependence

Previous studies of Barkhausen noise in PZT have been limited to the energy spectrum (slew rate response voltages versus time), showing agreement with avalanche models; in barium titanate other exponents have been measured acoustically, but only at ambient temperatures. In the present study we report the Omori exponent (0.95 [Formula: see text] 0.03) for aftershocks in PZT and extend the barium titanate studies to a wider range of temperature.

Multifractal analysis of financial markets: a review

Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.

Physics and financial economics (1776-2014): puzzles, Ising and agent-based models

This short review presents a selected history of the mutual fertilization between physics and economics--from Isaac Newton and Adam Smith to the present. The fundamentally different perspectives embraced in theories developed in financial economics compared with physics are dissected with the examples of the volatility smile and of the excess volatility puzzle. The role of the Ising model of phase transitions to model social and financial systems is reviewed, with the concepts of random utilities and the logit model as the analog of the Boltzmann factor in statistical physics. Recent extensions in terms of quantum decision theory are also covered. A wealth of models are discussed briefly that build on the Ising model and generalize it to account for the many stylized facts of financial markets. A summary of the relevance of the Ising model and its extensions is provided to account for financial bubbles and crashes. The review would be incomplete if it did not cover the dynamical field of agent-based models (ABMs), also known as computational economic models, of which the Ising-type models are just special ABM implementations. We formulate the 'Emerging Intelligence Market Hypothesis' to reconcile the pervasive presence of 'noise traders' with the near efficiency of financial markets. Finally, we note that evolutionary biology, more than physics, is now playing a growing role to inspire models of financial markets.

Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock T{c} to be the time for which the market volatility V(T{c}) has a peak that exceeds a predetermined threshold. The cascade of high volatility "aftershocks" triggered by the "main shock" is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws-the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001-2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude M≡log{10} V(T{c}) and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on M , both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Epileptic seizures: Quakes of the brain?

A dynamical analogy supported by five scale-free statistics (the Gutenberg-Richter distribution of event sizes, the distribution of interevent intervals, the Omori and inverse Omori laws, and the conditional waiting time until the next event) is shown to exist between two classes of seizures ("focal" in humans and generalized in animals) and earthquakes. Increments in excitatory interneuronal coupling in animals expose the system's dependence on this parameter and its dynamical transmutability: moderate increases lead to power-law behavior of seizure energy and interevent times, while marked ones to scale-free (power-law) coextensive with characteristic scales and events. The coextensivity of power law and characteristic size regimes is predicted by models of coupled heterogeneous threshold oscillators of relaxation and underscores the role of coupling strength in shaping the dynamics of these systems.

Limits of declustering methods for disentangling exogenous from endogenous events in time series with foreshocks, main shocks, and aftershocks

Many time series in natural and social sciences can be seen as resulting from an interplay between exogenous influences and an endogenous organization. We use a simple epidemic-type aftershock model of events occurring sequentially, in which future events are influenced (partially triggered) by past events to ask the question of how well can one disentangle the exogenous events from the endogenous ones. We apply both model-dependent and model-independent stochastic declustering methods to reconstruct the tree of ancestry and estimate key parameters. In contrast with previously reported positive results, we have to conclude that declustered catalogs are rather unreliable for the synthetic catalogs that we have investigated, which contains of the order of thousands of events, typical of realistic applications. The estimated rates of exogenous events suffer from large errors. The branching ratio n, quantifying the fraction of events that have been triggered by previous events, is also badly estimated in general from declustered catalogs. We find, however, that the errors tend to be smaller and perhaps acceptable in some cases for small triggering efficiency and branching ratios. The high level of randomness together with the long memory makes the stochastic reconstruction of trees of ancestry and the estimation of the key parameters perhaps intrinsically unreliable for long-memory processes. For shorter memories (larger "bare" Omori exponent), the results improve significantly.

Generic multifractality in exponentials of long memory processes

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent phi+1/2, where phi>0. This generalizes previous studies performed only with phi=0(with a truncation at an integral scale) by showing that multifractality holds over a remarkably large range of dimensionless scales for phi>0. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation phi from 1/2 and of another parameter sigma2 embodying information on the short-range amplitude of the memory kernel, the ultraviolet cutoff ("viscous") scale, and the variance of the white-noise innovations. In these processes, both a viscous scale and an integral scale naturally appear, bracketing the "inertial" scaling regime. We exhibit a surprisingly good collapse of the multifractal spectra zeta(q) on a universal scaling function, which enables us to derive high-order multifractal exponents from the small-order values and also obtain a given multifractal spectrum zeta(q) by different combinations of phi and sigma2.

Vere-Jones' self-similar branching model

Motivated by its potential application to earthquake statistics as well as for its intrinsic interest in the theory of branching processes, we study the exactly self-similar branching process introduced recently by Vere-Jones. This model extends the ETAS class of conditional self-excited branching point-processes of triggered seismicity by removing the problematic need for a minimum (as well as maximum) earthquake size. To make the theory convergent without the need for the usual ultraviolet and infrared cutoffs, the distribution of magnitudes m' of daughters of first-generation of a mother of magnitude m has two branches m < m' with exponent beta - d and m' > m with exponent beta + d, where beta and d are two positive parameters. We investigate the condition and nature of the subcritical, critical, and supercritical regime in this and in an extended version interpolating smoothly between several models. We predict that the distribution of magnitudes of events triggered by a mother of magnitude m over all generations has also two branches m' < m with exponent and with exponent beta - h, with h=d squareroot of (1-s), where s is the fraction of triggered events. This corresponds to a renormalization of the exponent d into h by the hierarchy of successive generations of triggered events. For a significant part of the parameter space, the distribution of magnitudes over a full catalog summed over an average steady flow of spontaneous sources (immigrants) reproduces the distribution of the spontaneous sources with a single branch and is blind to the exponents beta, d of the distribution of triggered events. Since the distribution of earthquake magnitudes is usually obtained with catalogs including many sequences, we conclude that the two branches of the distribution of aftershocks are not directly observable and the model is compatible with real seismic catalogs. In summary, the exactly self-similar Vere-Jones model provides an attractive new approach to model triggered seismicity, which alleviates delicate questions on the role of magnitude cutoffs in other non-self-similar models. The new prediction concerning two branches in the distribution of magnitudes of aftershocks could be tested with recently introduced stochastic reconstruction methods, tailored to disentangle the different triggered sequences.