arXiv subjects

A. Saichev

Publications and source records attributed to A. Saichev.

At least 19 recordsLinked to original sources

arXiv2017

Statistics of seismic cluster durations

Using the standard ETAS model of triggered seismicity, we present a rigorous theoretical analysis of the main statistical properties of temporal clusters, defined as the group of events triggered by a given main shock of fixed magnitude m that occurred at the origin of time, at times larger than some present time t. Using the technology of generating probability function (GPF), we derive the explicit expressions for the GPF of the number of future offsprings in a given temporal seismic cluster, defining, in particular, the statistics of the cluster's duration and the cluster's offsprings maximal magnitudes. We find the remarkable result that the magnitude difference between the largest and second largest event in the future temporal cluster is distributed according to the regular Gutenberg-Richer law that controls the unconditional distribution of earthquake magnitudes. For earthquakes obeying the Omori-Utsu law for the distribution of waiting times between triggering and triggered events, we show that the distribution of the durations of temporal clusters of events of magnitudes above some detection threshold νhas a power law tail that is fatter in the non-critical regime $n<1$ than in the critical case n=1. This paradoxical behavior can be rationalised from the fact that generations of all orders cascade very fast in the critical regime and accelerate the temporal decay of the cluster dynamics.

physics.geo-ph↗

arXiv2014

Super-exponential endogenous bubbles in an equilibrium model of rational and noise traders

We introduce a model of super-exponential financial bubbles with two assets (risky and risk-free), in which rational investors and noise traders co-exist. Rational investors form expectations on the return and risk of a risky asset and maximize their constant relative risk aversion expected utility with respect to their allocation on the risky asset versus the risk-free asset. Noise traders are subjected to social imitation and follow momentum trading. Allowing for random time-varying herding propensity, we are able to reproduce several well-known stylized facts of financial markets such as a fat-tail distribution of returns and volatility clustering. In particular, we observe transient faster-than-exponential bubble growth with approximate log-periodic behavior and give analytical arguments why this follows from our framework. The model accounts well for the behavior of traders and for the price dynamics that developed during the dotcom bubble in 1995-2000. Momentum strategies are shown to be transiently profitable, supporting these strategies as enhancing herding behavior.

q-fin.ST↗

arXiv2013

Super-linear scaling of offsprings at criticality in branching processes

For any branching process, we demonstrate that the typical total number $r_{\rm mp}(ντ)$ of events triggered over all generations within any sufficiently large time window $τ$ exhibits, at criticality, a super-linear dependence $r_{\rm mp}(ντ) \sim (ντ)^γ$ (with $γ>1$) on the total number $ντ$ of the immigrants arriving at the Poisson rate $ν$. In branching processes in which immigrants (or sources) are characterized by fertilities distributed according to an asymptotic power law tail with tail exponent $1 < γ\leqslant 2$, the exponent of the super-linear law for $r_{\rm mp}(ντ)$ is identical to the exponent $γ$ of the distribution of fertilities. For $γ>2$ and for standard branching processes without power law distribution of fertilities, $r_{\rm mp}(ντ) \sim (ντ)^2$. This novel scaling law replaces and tames the divergence $ντ/(1-n)$ of the mean total number ${\bar R}_t(τ)$ of events, as the branching ratio (defined as the average number of triggered events of first generation per source) tends to 1. The derivation uses the formalism of generating probability functions. The corresponding prediction is confirmed by numerical calculations and an heuristic derivation enlightens its underlying mechanism. We also show that ${\bar R}_t(τ)$ is always linear in $ντ$ even at criticality ($n=1$). Our results thus illustrate the fundamental difference between the mean total number, which is controlled by a few extremely rare realizations, and the typical behavior represented by $r_{\rm mp}(ντ)$.

physics.data-an↗

arXiv2012

Fertility Heterogeneity as a Mechanism for Power Law Distributions of Recurrence Times

We study the statistical properties of recurrence times in the self-excited Hawkes conditional Poisson process, the simplest extension of the Poisson process that takes into account how the past events influence the occurrence of future events. Specifically, we analyze the impact of the power law distribution of fertilities with exponent α, where the fertility of an event is the number of aftershocks of first generation that it triggers, on the probability distribution function (pdf) f(τ) of the recurrence times τbetween successive events. The other input of the model is an exponential Omori law quantifying the pdf of waiting times between an event and its first generation aftershocks, whose characteristic time scale is taken as our time unit. At short time scales, we discover two intermediate power law asymptotics, f(τ) ~ τ^{-(2-α)} for τ<< τ_c and f(τ) ~ τ^{-α} for τ_c << τ<< 1, where τ_c is associated with the self-excited cascades of aftershocks. For 1 << τ<< 1/ν, we find a constant plateau f(τ) ~ const, while at long times, 1/ν< τ, f(τ) ~ e^{-ντ} has an exponential tail controlled by the arrival rate νof exogenous events. These results demonstrate a novel mechanism for the generation of power laws in the distribution of recurrence times, which results from a power law distribution of fertilities in the presence of self-excitation and cascades of triggering.

physics.geo-ph↗

arXiv2012

A simple microstructure return model explaining microstructure noise and Epps effects

We present a simple microstructure model of financial returns that combines (i) the well-known ARFIMA process applied to tick-by-tick returns, (ii) the bid-ask bounce effect, (iii) the fat tail structure of the distribution of returns and (iv) the non-Poissonian statistics of inter-trade intervals. This model allows us to explain both qualitatively and quantitatively important stylized facts observed in the statistics of microstructure returns, including the short-ranged correlation of returns, the long-ranged correlations of absolute returns, the microstructure noise and Epps effects. According to the microstructure noise effect, volatility is a decreasing function of the time scale used to estimate it. Paradoxically, the Epps effect states that cross correlations between asset returns are increasing functions of the time scale at which the returns are estimated. The microstructure noise is explained as the result of the negative return correlations inherent in the definition of the bid-ask bounce component (ii). In the presence of a genuine correlation between the returns of two assets, the Epps effect is due to an average statistical overlap of the momentum of the returns of the two assets defined over a finite time scale in the presence of the long memory process (i).

q-fin.TR↗

arXiv2011

Time-Bridge Estimators of Integrated Variance

We present a set of log-price integrated variance estimators, equal to the sum of open-high-low-close bridge estimators of spot variances within $n$ subsequent time-step intervals. The main characteristics of some of the introduced estimators is to take into account the information on the occurrence times of the high and low values. The use of the high's and low's of the bridge associated with the original process makes the estimators significantly more efficient that the standard realized variance estimators and its generalizations. Adding the information on the occurrence times of the high and low values improves further the efficiency of the estimators, much above those of the well-known realized variance estimator and those derived from the sum of Garman and Klass spot variance estimators. The exact analytical results are derived for the case where the underlying log-price process is an Itô stochastic process. Our results suggests more efficient ways to record financial prices at intermediate frequencies.

q-fin.ST↗

arXiv2011

Generating Functions and Stability Study of Multivariate Self-Excited Epidemic Processes

We present a stability study of the class of multivariate self-excited Hawkes point processes, that can model natural and social systems, including earthquakes, epileptic seizures and the dynamics of neuron assemblies, bursts of exchanges in social communities, interactions between Internet bloggers, bank network fragility and cascading of failures, national sovereign default contagion, and so on. We present the general theory of multivariate generating functions to derive the number of events over all generations of various types that are triggered by a mother event of a given type. We obtain the stability domains of various systems, as a function of the topological structure of the mutual excitations across different event types. We find that mutual triggering tends to provide a significant extension of the stability (or subcritical) domain compared with the case where event types are decoupled, that is, when an event of a given type can only trigger events of the same type.

cond-mat.stat-mech↗

arXiv2011

Hierarchy of Temporal Responses of Multivariate Self-Excited Epidemic Processes

We present the first exact analysis of some of the temporal properties of multivariate self-excited Hawkes conditional Poisson processes, which constitute powerful representations of a large variety of systems with bursty events, for which past activity triggers future activity. The term "multivariate" refers to the property that events come in different types, with possibly different intra- and inter-triggering abilities. We develop the general formalism of the multivariate generating moment function for the cumulative number of first-generation and of all generation events triggered by a given mother event (the "shock") as a function of the current time $t$. This corresponds to studying the response function of the process. A variety of different systems have been analyzed. In particular, for systems in which triggering between events of different types proceeds through a one-dimension directed or symmetric chain of influence in type space, we report a novel hierarchy of intermediate asymptotic power law decays $\sim 1/t^{1-(m+1)θ}$ of the rate of triggered events as a function of the distance $m$ of the events to the initial shock in the type space, where $0 < θ<1$ for the relevant long-memory processes characterizing many natural and social systems. The richness of the generated time dynamics comes from the cascades of intermediate events of possibly different kinds, unfolding via a kind of inter-breeding genealogy.

cond-mat.stat-mech↗

arXiv2010

Zipf's law and maximum sustainable growth

Zipf's law states that the number of firms with size greater than S is inversely proportional to S. Most explanations start with Gibrat's rule of proportional growth but require additional constraints. We show that Gibrat's rule, at all firm levels, yields Zipf's law under a balance condition between the effective growth rate of incumbent firms (which includes their possible demise) and the growth rate of investments in entrant firms. Remarkably, Zipf's law is the signature of the long-term optimal allocation of resources that ensures the maximum sustainable growth rate of an economy.

physics.soc-ph↗

arXiv2009

Effects of Diversity and Procrastination in Priority Queuing Theory: the Different Power Law Regimes

Empirical analysis show that, after the update of a browser, the publication of the vulnerability of a software, or the discovery of a cyber worm, the fraction of computers still using the older version, or being not yet patched, or exhibiting worm activity decays as power laws $\sim 1/t^α$ with $0 < α\leq 1$ over time scales of years. We present a simple model for this persistence phenomenon framed within the standard priority queuing theory, of a target task which has the lowest priority compared with all other tasks that flow on the computer of an individual. We identify a "time deficit" control parameter $β$ and a bifurcation to a regime where there is a non-zero probability for the target task to never be completed. The distribution of waiting time ${\cal T}$ till the completion of the target task has the power law tail $\sim 1/t^{1/2}$, resulting from a first-passage solution of an equivalent Wiener process. Taking into account a diversity of time deficit parameters in a population of individuals, the power law tail is changed into $1/t^α$ with $α\in(0.5,\infty)$, including the well-known case $1/t$. We also study the effect of "procrastination", defined as the situation in which the target task may be postponed or delayed even after the individual has solved all other pending tasks. This new regime provides an explanation for even slower apparent decay and longer persistence.

physics.soc-ph↗

arXiv2009

Most Efficient Homogeneous Volatility Estimators

We present a comprehensive theory of homogeneous volatility (and variance) estimators of arbitrary stochastic processes that fully exploit the OHLC (open, high, low, close) prices. For this, we develop the theory of most efficient point-wise homogeneous OHLC volatility estimators, valid for any price processes. We introduce the "quasi-unbiased estimators", that can address any type of desirable constraints. The main tool of our theory is the parsimonious encoding of all the information contained in the OHLC prices for a given time interval in the form of the joint distributions of the high-minus-open, low-minus-open and close-minus-open values, whose analytical expression is derived exactly for Wiener processes with drift. The distributions can be calculated to yield the most efficient estimators associated with any statistical properties of the underlying log-price stochastic process. Applied to Wiener processes for log-prices with drift, we provide explicit analytical expressions for the most efficient point-wise volatility and variance estimators, based on the analytical expression of the joint distribution of the high-minus-open, low-minus-open and close-minus-open values. The efficiency of the new proposed estimators is favorably compared with that of the Garman-Klass, Roger-Satchell and maximum likelihood estimators.

q-fin.ST↗

arXiv2009

Generation-by-Generation Dissection of the Response Function in Long Memory Epidemic Processes

In a number of natural and social systems, the response to an exogenous shock relaxes back to the average level according to a long-memory kernel $\sim 1/t^{1+θ}$ with $0 \leq θ<1$. In the presence of an epidemic-like process of triggered shocks developing in a cascade of generations at or close to criticality, this "bare" kernel is renormalized into an even slower decaying response function $\sim 1/t^{1-θ}$. Surprisingly, this means that the shorter the memory of the bare kernel (the larger $1+θ$), the longer the memory of the response function (the smaller $1-θ$). Here, we present a detailed investigation of this paradoxical behavior based on a generation-by-generation decomposition of the total response function, the use of Laplace transforms and of "anomalous" scaling arguments. The paradox is explained by the fact that the number of triggered generations grows anomalously with time at $\sim t^θ$ so that the contributions of active generations up to time $t$ more than compensate the shorter memory associated with a larger exponent $θ$. This anomalous scaling results fundamentally from the property that the expected waiting time is infinite for $0 \leq θ\leq 1$. The techniques developed here are also applied to the case $θ>1$ and we find in this case that the total renormalized response is a {\bf constant} for $t < 1/(1-n)$ followed by a cross-over to $\sim 1/t^{1+θ}$ for $t \gg 1/(1-n)$.

physics.data-an↗

arXiv2008

Theory of Zipf's Law and of General Power Law Distributions with Gibrat's law of Proportional Growth

We summarize a book under publication with his title written by the three present authors, on the theory of Zipf's law, and more generally of power laws, driven by the mechanism of proportional growth. The preprint is available upon request from the authors. For clarity, consistence of language and conciseness, we discuss the origin and conditions of the validity of Zipf's law using the terminology of firms' asset values. We use firms at the entities whose size distributions are to be explained. It should be noted, however, that most of the relations discussed in this book, especially the intimate connection between Zipf's and Gilbrat's laws, underlie Zipf's law in diverse scientific areas. The same models and variations thereof can be straightforwardly applied to any of the other domains of application.

q-fin.GN↗

arXiv2007

Nonlinear theory and tests of earthquake recurrence times

We develop an efficient numerical scheme to solve accurately the set of nonlinear integral equations derived previously in (Saichev and Sornette, 2007), which describes the distribution of inter-event times in the framework of a general model of earthquake clustering with long memory. Detailed comparisons between the linear and nonlinear versions of the theory and direct synthetic catalogs show that the nonlinear theory provides an excellent fit to the synthetic catalogs, while there are significant biases resulting from the use of the linear approximation. We then address the suggestions proposed by some authors to use the empirical distribution of inter-event times to obtain a better determination of the so-called clustering parameter. Our theory and tests against synthetic and empirical catalogs find a rather dramatic lack of power for the distribution of inter-event times to distinguish between quite different sets of parameters, casting doubt on the usefulness of this statistics for the specific purpose of identifying the clustering parameter.

physics.data-an↗

arXiv2006

Theory of Earthquake Recurrence Times

The statistics of recurrence times in broad areas have been reported to obey universal scaling laws, both for single homogeneous regions (Corral, 2003) and when averaged over multiple regions (Bak et al.,2002). These unified scaling laws are characterized by intermediate power law asymptotics. On the other hand, Molchan (2005) has presented a mathematical proof that, if such a universal law exists, it is necessarily an exponential, in obvious contradiction with the data. First, we generalize Molchan's argument to show that an approximate unified law can be found which is compatible with the empirical observations when incorporating the impact of the Omori law of earthquake triggering. We then develop the full theory of the statistics of inter-event times in the framework of the ETAS model of triggered seismicity and show that the empirical observations can be fully explained. Our theoretical expression fits well the empirical statistics over the whole range of recurrence times, accounting for different regimes by using only the physics of triggering quantified by Omori's law. The description of the statistics of recurrence times over multiple regions requires an additional subtle statistical derivation that maps the fractal geometry of earthquake epicenters onto the distribution of the average seismic rates in multiple regions. This yields a prediction in excellent agreement with the empirical data for reasonable values of the fractal dimension $d \approx 1.8$, the average clustering ratio $n \approx 0.9$, and the productivity exponent $α\approx 0.9$ times the $b$-value of the Gutenberg-Richter law.

physics.geo-ph↗

arXiv2006

"Universal" Distribution of Inter-Earthquake Times Explained

We propose a simple theory for the ``universal'' scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of inter-event times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforces the view that triggering of earthquakes by other earthquakes is a key physical mechanism to understand seismicity.

physics.geo-ph↗

arXiv2006

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 $ϕ+1/2$, where $ϕ>0$. This generalizes previous studies performed only with $ϕ=0$ (with a truncation at an integral scale), by showing that multifractality holds over a remarkably large range of dimensionless scales for $ϕ>0$. The intermittency multifractal coefficient can be tuned continuously as a function of the deviation $ϕ$ from 1/2 and of another parameter $σ^2$ embodying information on the short-range amplitude of the memory kernel, the ultra-violet cut-off (``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 $ζ(q)$ 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 $ζ(q)$ by different combinations of $ϕ$ and $σ^2$.

cond-mat.stat-mech↗

arXiv2005

Renormalization of the ETAS branching model of triggered seismicity from total to observable seismicity

Several recent works point out that the crowd of small unobservable earthquakes (with magnitudes below the detection threshold $m_d$) may play a significant and perhaps dominant role in triggering future seismicity. Using the ETAS branching model of triggered seismicity, we apply the formalism of generating probability functions to investigate how the statistical properties of observable earthquakes differ from the statistics of all events. The ETAS (epidemic-type aftershock sequence) model assumes that each earthquake can trigger other earthquakes (``aftershocks''). An aftershock sequence results in this model from the cascade of aftershocks of each past earthquake. The triggering efficiency of earthquakes is assumed to vanish below a lower magnitude limit $m_0$, in order to ensure the convergence of the theory and may reflect the physics of state-and-velocity frictional rupture. We show that, to a good approximation, the ETAS model is renormalized onto itself under what amounts to a decimation procedure $m_0 \to m_d$, with just a renormalization of the branching ratio from $n$ to an effective value $n(m_d)$. Our present analysis thus confirms, for the full statistical properties, the results obtained previously by one of us and Werner, based solely on the average seismic rates (the first-order moment of the statistics). However, our analysis also demonstrates that this renormalization is not exact, as there are small corrections which can be systematically calculated, in terms of additional contributions that can be mapped onto a different branching model (a new relevant direction in the language of the renormalization group).

physics.geo-ph↗