We analyze the probability density function (PDF) of waiting times between financial loss exceedances. The empirical PDFs are fitted with the self-excited Hawkes conditional Poisson process with a long power law memory kernel. The Hawkes process is the simplest extension of the Poisson process that takes into account h…
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48 results for “power Poisson laws”
We develop dependent hierarchical normalized random measures and apply them to dynamic topic modeling. The dependency arises via superposition, subsampling and point transition on the underlying Poisson processes of these measures. The measures used include normalised generalised Gamma processes that demonstrate power …
We introduce solvable stochastic dealer models, which can reproduce basic empirical laws of financial markets such as the power law of price change. Starting from the simplest model that is almost equivalent to a Poisson random noise generator, the model becomes fairly realistic by adding only two effects, the self-mod…
Study models market volatility with persistent and temporary impacts.
problem Microstructure of rough volatility models driven by Poisson measures.
method Existence and uniqueness of solutions for stochastic path-dependent Volterra equations.
result Volatility process converges to fractional Heston model with spikes.
Proposes a test for stochastic block models with bounded degrees.
problem Testing Erdös-Rényi model versus bisection stochastic block model with bounded degrees.
method Likelihood-ratio (LR) type procedure based on regularization.
result Limit distributions as power Poisson laws under null and alternative hypotheses.
The paper analyzes convergence rates for stochastic approximation and reinforcement learning.
problem Establishing almost sure convergence rates for stochastic approximation and reinforcement learning under Markovian noise.
method A novel Lyapunov drift construction that applies a Poisson-equation based correction for Markovian noise to the Moreau-envelope smoothing for contractive mappings.
result Almost sure convergence rates for specific learning rates are derived, with rates arbitrarily close to and .
We analyze waiting times for price changes in a foreign currency exchange rate. Recent empirical studies of high frequency financial data support that trades in financial markets do not follow a Poisson process and the waiting times between trades are not exponentially distributed. Here we show that our data is well ap…
Invariant reduction preserves Poisson structures in PDEs.
problem Preserving Poisson structures in invariant solutions of PDEs.
method Invariant reduction applied to PDEs through Hamiltonian operators and Poisson bivectors.
result Inherited Poisson brackets match original systems up to sign.
Market lawsq-fin.ST
Power laws detected in financial data, modeled with random multipliers.
problem Detecting power laws in financial data.
method Investigated data from financial instruments, proposed a model based on sums of Maxwell-Boltzmann distributions with random multipliers.
result Detected power laws with various exponents in financial data, proposed a universal model.
Higher-order conservation laws for the non-linear Poisson equation via the characteristic cohomologymath.DG
We study higher-order conservation laws of the non-linearizable elliptic Poisson equation as elements of the characteristic cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form for diffe…
New SDE model from machine learning optimization with unique stationary distribution.
problem Stationary distribution of machine learning optimization models.
method Proved ergodicity and unique stationary distribution of power-law dynamic SDE.
result Power-law dynamic has a unique stationary distribution and is ergodic.
This work analyzes neural scaling laws using power-law data spectra and derives analytical expressions for generalization error.
problem Understanding how neural network performance scales with key factors like data size and model complexity.
method Statistical mechanics techniques applied to one-pass stochastic gradient descent in a student-teacher framework.
result Derivation of analytical expressions for generalization error under power-law data spectra and identification of conditions for power-law scaling.
The inception of Symplectic Geometry: the works of Lagrange and Poisson during the years 1808-1810math.HO
The concept of a symplectic structure first appeared in the works of Lagrange on the so-called "method of variation of the constants". These works are presented, together with those of Poisson, who first defined the composition law called today the "Poisson bracket". The method of variation of the constants is presente…
Superposition accelerates training to a universal power-law exponent.
problem Training dynamics in neural networks.
method Teacher-student framework and analytic theory.
result Superposition leads to a universal power-law exponent of ~1, independent of data and channel statistics.
Learning curves theory for hierarchically compositional data with power-law distributed featuresstat.ML
Unified theory for neural scaling laws in hierarchically compositional data.
problem Understanding neural scaling laws in hierarchically compositional data.
method Probabilistic context-free grammars and power-law distributed production rules.
result Unified learning curve behavior for classification and next-token prediction tasks.
Develops a simple model to understand learning curves for arbitrary power laws.
problem Lack of theoretical understanding of scaling laws in machine learning.
method Analyzes a toy model to determine if learning curves are universal or depend on data distribution.
result Determines that learning curves can exhibit for arbitrary power .
Modelling financial markets by the multiplicative sequence of tradescond-mat.stat-mech
We introduce the stochastic multiplicative point process modelling trading activity of financial markets. Such a model system exhibits power-law spectral density S(f) ~ 1/f**beta, scaled as power of frequency for various values of beta between 0.5 and 2. Furthermore, we analyze the relation between the power-law autoco…
Study on KRR with power-law data, showing better sample complexity.
problem High-dimensional kernel ridge regression with anisotropic power-law covariance.
method Explicit characterization of kernel spectrum and asymptotic analysis of excess risk.
result Sample complexity is governed by effective dimension, not ambient dimension.
Large models follow power laws in performance with dataset size or parameters.
problem Understanding neural scaling laws in large language models.
method Joint generative data model and random feature model.
result Modeling and solving the dual limit reveals insights into scaling laws.
We use data on wealth of the richest persons taken from the "rich lists" provided by business magazines like Forbes to verify if upper tails of wealth distributions follow, as often claimed, a power-law behaviour. The data sets used cover the world's richest persons over 1996-2012, the richest Americans over 1988-2012,…
An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is …
A mechanism to derive multi-power law functions: an application in the econophysics frameworkphysics.data-an
It is generally recognized that economical systems, and more in general complex systems, are characterized by power law distributions. Sometime, these distributions show a changing of the slope in the tail so that, more appropriately, they show a multi-power law behavior. We present a method to derive analytically a tw…
Power law relaxation in a complex system: Omori law after a financial market crashcond-mat.stat-mech
We study the relaxation dynamics of a financial market just after the occurrence of a crash by investigating the number of times the absolute value of an index return is exceeding a given threshold value. We show that the empirical observation of a power law evolution of the number of events exceeding the selected thre…
Resemblance of the power-law scaling behavior of a non-Markovian and nonlinear point processescond-mat.stat-mech
Non-Markovian point process shows power-law scaling, similar to nonlinear Markovian process.
problem Understanding the scaling behavior of non-Markovian point processes.
method Analyzed a confined fractional Brownian motion-driven point process and compared it to a nonlinear Markovian process.
result A nonlinear Markovian process can reproduce the power-law scaling behavior of a non-Markovian point process.
Study non-integer power-law potentials for Schrödinger operators using Lie-Rinehart algebras.
problem Analyzing Schrödinger operators with non-integer power-law potentials.
method Using Lie-Rinehart algebras and microlocal analysis.
result Microlocal analysis can be applied to Schrödinger operators with non-integer power-law potentials.
The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. Th…
Auto-regressive conditionally heteroskedastic (ARCH) family models are still used, by practitioners in business and economic policy making, as a conditional volatility forecasting models. Furthermore ARCH models still are attracting an interest of the researchers. In this contribution we consider the well known GARCH(1…
We show that there is a common mode of origin for the power laws observed in two different models: (i) the Pareto law for the distribution of money among the agents with random saving propensities in an ideal gas-like market model and (ii) the Gutenberg-Richter law for the distribution of overlaps in a fractal-overlap …
Limit order placement as an utility maximization problem and the origin of power law distribution of limit order pricesphysics.soc-ph
I consider the problem of the optimal limit order price of a financial asset in the framework of the maximization of the utility function of the investor. The analytical solution of the problem gives insight on the origin of the recently empirically observed power law distribution of limit order prices. In the framewor…
Ubiquitous power law scaling in nonlinear self-excited Hawkes processescond-mat.stat-mech
New mechanism found for power laws including Zipf's law.
problem Understanding the ubiquity of power law distributions.
method Introduced nonlinear self-excited Hawkes processes with fast-accelerating intensities.
result Wide class of nonlinear Hawkes processes have power law intensity PDFs.
The study finds significant power-law cross correlations in Bitcoin's return-volatility dynamics.
problem Investigating asymmetry in Bitcoin's return-volatility relationships.
method Analysis of daily and high-frequency Bitcoin data to identify cross correlations.
result Power-law cross correlations between returns and future volatilities are observed, indicating long-range dependencies.
Maps with a single face converge to hyperbolic surfaces in large genus.
problem Understanding geometric properties of high genus maps.
method Analyzing uniformly random maps and their convergence to hyperbolic surfaces.
result Lengths of simple cycles converge to a Poisson process.
On the interplay between short and long term memory in the power-law cross-correlations settingstat.ME
We focus on emergence of the power-law cross-correlations from processes with both short and long term memory properties. In the case of correlated error-terms, the power-law decay of the cross-correlation function comes automatically with the characteristics of separate processes. Bivariate Hurst exponent is then equa…
We present a model for random simple graphs with a degree distribution that obeys a power law (i.e., is heavy-tailed). To attain this behavior, the edge probabilities in the graph are constructed from Bertoin-Fujita-Roynette-Yor (BFRY) random variables, which have been recently utilized in Bayesian statistics for the c…
The Power-law Tail Exponent of Income Distributionsphysics.soc-ph
In this paper we tackle the problem of estimating the power-law tail exponent of income distributions by using the Hill's estimator. A subsample semi-parametric bootstrap procedure minimising the mean squared error is used to choose the power-law cutoff value optimally. This technique is applied to personal income data…
The paper presents a multi-power law for predicting loss curves across different learning rate schedules.
problem Understanding and optimizing the relationship between model performance and hyperparameters, especially learning rates.
method Proposes a multi-power law that combines power laws based on the sum of learning rates and additional laws for loss reduction due to decay.
result The multi-power law accurately predicts loss curves for unseen learning rate schedules and finds a schedule that outperforms cosine learning rate.
LLMs learn peaked distributions slowly due to power-law losses.
problem Slow convergence of loss in training large language models.
method Systematic analysis of toy models and empirical evaluation of LLMs.
result Power-law time scaling with an exponent of 1/3 for learning peaked distributions.
Paper proves SVV model reproduces power-law skew in implied volatilities.
problem Reproducing power-law behavior in implied volatility skew.
method Analytical proof using Malliavin calculus and Volterra kernel selection.
result SVV model reproduces power-law skew under correct kernel choice.
We introduce preferential behavior into the study on statistical mechanics of money circulation. The computer simulation results show that the preferential behavior can lead to power laws on distributions over both holding time and amount of money held by agents. However, some constraints are needed in generation mecha…
On the origin of power law tails in price fluctuationscond-mat.stat-mech
In a recent Nature paper, Gabaix et al. \cite{Gabaix03} presented a theory to explain the power law tail of price fluctuations. The main points of their theory are that volume fluctuations, which have a power law tail with exponent roughly -1.5, are modulated by the average market impact function, which describes the r…
One of the first steps to understand and forecast economic downturns is identifying their frequency distribution, but it remains uncertain. This problem is common in phenomena displaying power-law-like distributions. Power laws play a central role in complex systems theory; therefore, the current limitations in the ide…
On the Origin of Power-Law Fluctuations in Stock Pricescond-mat.dis-nn
We respond to the issues discussed by Farmer and Lillo (FL) related to our proposed approach to understanding the origin of power-law distributions in stock price fluctuations. First, we extend our previous analysis to 1000 US stocks and perform a new estimation of market impact that accounts for splitting of large ord…
Volatility has to be roughq-fin.MF
Volatility models must be rough to match market skew.
problem Inconsistent non-rough volatility models with power law volatility skew.
method Asymptotic expansion and continuous price dynamics analysis.
result Volatility must be rough to align with market skew.
We introduce a new statistical tool (the TP-statistic and TE-statistic) designed specifically to compare the behavior of the sample tail of distributions with power-law and exponential tails as a function of the lower threshold u. One important property of these statistics is that they converge to zero for power laws o…
Study uses OT to simulate markets, revealing power-law returns are driven by informational effect.
problem Reproduce power-law returns in financial markets using realistic simulations.
method Constructed artificial markets, used optimal transport (OT) to measure similarity, incrementally introduced behavioral components.
result Informational effect of prices is dominant in reproducing power-law returns, and multiple components interact synergistically.
This study examines evolving networks of P2P lending relationships, revealing scale-free characteristics and the impact of interest rate and term.
problem Understanding the structural characteristics of evolving networks of debtor-creditor relationships in P2P lending.
method Modeling P2P lending networks as evolving networks with addition and deletion of nodes, analyzing attributes and factors affecting the scale-free exponent.
result P2P lending networks are scale-free with significant influence from interest rate and term on the exponent of power-law.
We study by theoretical analysis and by direct numerical simulation the dynamics of a wide class of asynchronous stochastic systems composed of many autocatalytic degrees of freedom. We describe the generic emergence of truncated power laws in the size distribution of their individual elements. The exponents of the…
Ormerod and Mounfield analysed GDP data of 17 leading capitalist economies from 1870 to 1994 and concluded that the frequency of the duration of recessions is consistent with a power-law. But in fact the data is consistent with an exponential (Boltzmann-Gibbs) law.