Power-law Singularity Research Articles

Vallée Poussin sums of rational Fourier–Chebyshev integral operators and approximations of the Markov function

Rational approximations of the Markov function on the segment [ − 1 , 1 ] [-1,1] are studied. The Vallée Poussin sums of rational integral Fourier–Chebyshev operators as an approximation apparatus with a fixed number of geometrically distinct poles are chosen. For the constructed rational approximation method, integral representations of approximations and upper estimates of uniform approximations are established. For the Markov function with a measure whose derivative is a function that has a power-law singularity on the segment [ − 1 , 1 ] [-1, 1] , upper estimates of pointwise and uniform approximations and an asymptotic expression of a majorant of uniform approximations are found. The values of the parameters of the approximating function are determined at which the best uniform rational approximations are provided by this method. It is shown that in this case they have a higher rate of decrease in comparison with the corresponding polynomial analogs. As a corollary, rational approximations on a segment by Vallée Poussin sums of some elementary functions representable by a Markov function are considered.

Stock market volatility and multi-scale positive and negative bubbles

We study whether booms and busts in the stock market of the United States (US) drives its volatility. Given this, first, we employ the Multi-Scale Log-Periodic Power Law Singularity Confidence Indicator (MS-LPPLS-CI) approach to identify both positive and negative bubbles in the short-, medium, and long-term. We successfully detect major crashes and rallies during the weekly period from January 1973 to December 2020. Second, we utilize a nonparametric causality-in-quantiles approach to analyze the predictive impact of our bubble indicators on daily data-based weekly realized volatility (RV). This econometric framework allows us to circumvent potential misspecification due to nonlinearity and instability, rendering the results of weak causal influence derived from a linear framework invalid. The MS-LPPLS-CIs reveal strong evidence of predictability for RV over its entire conditional distribution. We observe relatively stronger impacts for the positive bubbles indicators, with our findings being robust to an alternative metric of volatility, namely squared returns, and weekly realized volatilities derived from 5 (RV5)- and 10 (RV10)-minutes interval intraday data. Furthermore, we detect evidence of predictability for RV5 and RV10 of nine other developed and emerging stock markets. In addition, we also find strong evidence of causal feedbacks from RV5 and RV10 on to the MS-LPPLS-CIs of the 10 countries considered. Finally, time-varying connectedness of the RVs of the G7 stock markets is also shown to be strongly (positively) predicted by the connectedness of the six bubbles indicators. Our findings have significant implications for investors and policymakers.

Detecting market bubbles: A generalized LPPLS neural network model

To enhance bubble detection capabilities, we introduce two significant improvements to the Log-Periodic Power Law Singularity (LPPLS) model: (1) a novel fitting approach, which yields more accurate predictions of critical price distributions within a single sample window; (2) a restructured neural network approach further enhances the estimations of the probability distributions of the critical points across both time and price dimensions, and it can be fine-tuned with real-world data. The simulation and practical applications to typical asset price bubbles in cryptocurrencies, commodities, and equity indices demonstrate that our refined model, the Generalized-LPPLS Neural Network (G-LPPLS-NN), outperforms all other models we examined in terms of predictive accuracy for critical point distributions.

One-dimensional Fermi polaron after a kick: Two-sided singularity of the momentum distribution, Bragg reflection and other exact results

A mobile impurity particle immersed in a quantum fluid forms a polaron - a quasiparticle consisting of the impurity and a local disturbance of the fluid around it. We ask what happens to a one-dimensional polaron after a kick, i.e. an abrupt application of a force that instantly delivers a finite impulse to the impurity. In the framework of an integrable model describing an impurity in a one-dimensional gas of fermions or hard-core bosons, we calculate the distribution of the polaron momentum established when the post-kick relaxation is over. A remarkable feature of this distribution is a two-sided power-law singularity. It emerges due to one of two processes. In the first process, the whole impulse is transferred to the polaron, without creating phonon-like excitations of the fluid. In the second process, the impulse is shared between the polaron and the center-of-mass motion of the fluid, again without creating any fluid excitations. The latter process is, in fact, a Bragg reflection at the edge of the emergent Brillouin zone. We carefully analyze the conditions for each of the two processes. The asymptotic form of the distribution in the vicinity of the singularity is derived.

Identifying price bubbles in global carbon markets: Evidence from the SADF test, GSADF test and LPPLS method

Amid growing climate concerns, Emissions Trading Schemes (ETS) serve as key mechanisms for reducing greenhouse gas emissions through market forces. This study delves into the phenomenon of price bubbles within six major ETSs globally: the European Union Emissions Trading Scheme (EU ETS), the New Zealand Emissions Trading Scheme (NZ ETS), the California-Québec Emissions Trading Scheme (CQ ETS), the South Korea Emissions Trading Schemes (K-ETS), the China Emissions Trading Scheme (C-ETS), and the United Kingdom Emissions Trading Schemes (UK ETS). Employing the sup augmented Dickey-Fuller (SADF) test, generalized sup ADF (GSADF) test, and log-periodic power law singularity (LPPLS) model, our analysis aims to uncover the dynamics and implications of these bubbles. Our findings reveal varied patterns of price bubbles across the schemes, with the EU ETS showing the highest frequency and the NZ ETS exhibiting the longest durations. The period of 2021 and 2022 marked a significant increase in bubble occurrences across all ETSs, suggesting the influence of post-COVID-19 economic recovery and geopolitical tensions on market stability. These bubbles are attributed to factors including carbon market policies, macroeconomic conditions, energy price fluctuations, and market uncertainties. The study offers valuable insights for compliance firms, policymakers, and investors on navigating carbon market volatility. Understanding bubble dynamics can aid in formulating more effective emission reduction strategies, carbon pricing policies, and investment decisions. This research underscores the importance of integrating monetary policy considerations with environmental objectives within ETS frameworks, paving the way for future research on predictive bubble detection and the development of robust carbon market regulations.

Does every cloud (bubble) have a silver lining? An investigation of ESG financial markets

This paper investigates the presence of financial bubbles in the environmentally friendly investments captured by the ESG markets. By using the log-periodic power law singularity framework, we identified several periods of positive and negative bubbles in the short, medium, and long term. Moreover, we examined the relationship between ESG attention sentiment and financial bubbles. We found an asymmetric effect of ESG sentiment on financial bubbles, i.e., increasing positive and decreasing negative bubbles. Our empirical results provide valuable insights into the stability of environmentally friendly markets, which help risk managers and policymakers respond appropriately to financial and social bubbles.

Monetary policy uncertainty and the price bubbles in energy markets

We examine the relationship between the monetary policy uncertainty (MPU) and the price bubbles in U.S. oil futures, including WTI crude oil future, heating oil future, and gasoline future. The Log Periodic Power Law Singularity (LPPLS) model is firstly used to analyze and validate the price bubbles of U.S. oil futures. We find that (1) the timing of local peaks of MPU closely aligns with the occurrence of price bubbles in U.S. oil futures, (2) MPU is significantly correlated with crude oil future price bubble but not with heating oil future (or gasoline future) price bubble, and MPU significantly amplify the magnitude of price bubble risk for crude oil future, (3) MPU is an effective set of factors for machine-learning based identification of all three U.S. oil future price bubbles in the non-linear model, even though the impact of MPU on heating oil (or gasoline) future price bubble is not significant in logit regression as the linear model. Collectively, our findings highlight the significance of MPU in relation to price bubbles in U.S. oil futures, offering new insights for investors and policymakers to explore the (non)-linear relationship between MPU and price bubbles in energy markets.

No reward—no effort: Will Bitcoin collapse near to the year 2140?

This paper explores whether the overall evolution of Bitcoin log-prices would manifest a log-period power-law singularity (LPPLS) signature, eventually resulting in the arrival of a finite-time singularity. Calibrating the LPPLS model using daily data on Bitcoin covering the 2011—2023 period, this study indeed finds evidence for a strong LPPLS signature suggesting the arrival of a spontaneous singularity in the year 2129. Further striking evidence suggests that Bitcoin will experience what we term a close-to-singularity-condition near to the year 2050—a remarkable coincidence with the recently documented arrival of a finite-time singularity in U.S. equities.

Detection of financial bubbles using a log‐periodic power law singularity (LPPLS) model

AbstractThis article provides a systematic review of the theoretical and empirical academic literature on the development and extension of the log‐periodic power law singularity (LPPLS) model, which is also known as the Johansen–Ledoit–Sornette (JLS) model or log‐periodic power law (LPPL) model. Developed at the interface of financial economics, behavioral finance and statistical physics, the LPPLS model provides a flexible and quantitative framework for detecting financial bubbles and crashes by capturing two salient empirical characteristics of price trajectories in speculative bubble regimes: the faster‐than‐exponential growth of price leading to unsustainable growth ending with a finite crash‐time and the accelerating log‐periodic oscillations. We also demonstrate the LPPLS model by detecting the recent bubble status of the S&P 500 index between April 2020 and December 2022, during which the S&P 500 index reaches its all‐time peak at the end of 2021. We find that the strong corrections of the S&P 500 index starting from January 2022 stem from the increasingly systemic instability of the stock market itself, while the well‐known external shocks, such as the decades‐high inflation, aggressive monetary policy tightening by the Federal Reserve, and the impact of the Russia/Ukraine war, may serve as sparks.This article is categorized under: Applications of Computational Statistics > Computational Finance Algorithms and Computational Methods > Computational Complexity Statistical Models > Nonlinear Models

Stock market bubbles and the realized volatility of oil price returns

Using monthly data for the G7 countries from 1973 to 2020, we study whether stock market bubbles help to forecast out-of-sample the realized volatility of oil price returns. We use the Multi-Scale Log-Periodic Power Law Singularity Confidence Indicator (MS-LPPLS-CI) approach to identify both positive and negative bubbles in the short-, medium, and long-term. First, we successfully detect major crashes and rallies using the MS-LPPLS-CIs. Having established the relevance of the bubbles indicators, and given the large number of them, we use widely-studied shrinkage (Lasso, elastic net, ridge regression) approaches to estimate our forecasting models. We find that stock market bubbles have predictive value for realized volatility at a short to intermediate forecast horizon. The number of bubble predictors included in the penalized forecasting models tend to increase in the forecast horizon. We obtain our main finding for the various types of stock market bubbles, and for good and bad realized volatilities.

Extreme events, economic uncertainty and speculation on occurrences of price bubbles in crude oil futures

This paper examines the impacts of extreme events, economic uncertainty and speculation on price bubbles in crude oil futures. For better forecast and estimate the positive/negative price bubbles of crude oil futures, the log-periodic power law singularity confidence multi-scale indicators application is used. The panel probit model examines impacts of extreme events, economic uncertainty and speculation on price bubbles in crude oil. The results show that the recent extreme events including Corona Virus Disease 2019 and Russia-Ukraine war significantly affect both positive/negative bubbles in crude oil futures. Measures of economic uncertainty significantly affect both positive/negative bubbles in the crude oil. However, the results find that the speculation significantly affect the positive bubbles and does not significantly affect the negative bubbles. Overall, extreme events, economic uncertainty, and speculation can all contribute to the occurrences of price bubbles in crude oil futures markets. This study provides new insights for the policy makers and market participants to monitor the occurrence of price bubbles.

On the Fourier asymptotics of absolutely continuous measures with power-law singularities

We prove sharp estimates on the time-average behavior of the squared absolute value of the Fourier transform of some absolutely continuous measures that may have power-law singularities, in the sense that their Radon–Nikodym derivatives diverge with a power-law order. We also discuss an application to spectral measures of finite-rank perturbations of the discrete Laplacian.

Detection of financial bubbles using a log-periodic power law singularity (LPPLS) model

Detection of financial bubbles using a log-periodic power law singularity (LPPLS) model

General Fractional Calculus Operators of Distributed Order

In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric Sonin kernels with power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann–Liouville and Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper.

Why topological data analysis detects financial bubbles?

We present a heuristic argument for the propensity of Topological Data Analysis (TDA) to detect early warning signals of critical transitions in financial time series. Our argument is based on the Log-Periodic Power Law Singularity (LPPLS) model, which characterizes financial bubbles as super-exponential growth (or decay) of an asset price superimposed with oscillations increasing in frequency and decreasing in amplitude when approaching a critical transition (tipping point). We show that whenever the LPPLS model is fitting with the data, TDA generates early warning signals. As an application, we illustrate this approach on a sample of positive and negative bubbles in the Bitcoin historical price.

A semi-analytical matrix formalism for stress singularities in anisotropic multi-material corners with frictional boundary and interface conditions

A new general semi-analytic procedure for the characterisation of singular asymptotic elastic states in the vicinity of the apex of linearly elastic anisotropic multi-material corners, including frictional contact, is developed and tested. The corners can consist of any finite number of homogeneous wedges defined by polar sectors. The variability of configurations covered is enormous as frictional contact can be considered on one or more outer boundary surfaces or interfaces, in addition to a large variety of homogeneous boundary conditions and perfect bonding or frictionless sliding interface conditions between wedges in the corner. The Coulomb rate-independent and dry frictional contact law is assumed. One of the novelties is that, in addition to the singularity exponent λ, the angle ω of the friction tangential stress vector on each frictional contact surface is an a priory unknown to be determined by solving a nonlinear corner eigensystem. The procedure, which considers power-law stress singularities, is based on the Stroh formalism of anisotropic elasticity, assuming generalised plane strain (2.5D) conditions, and on the semi-analytic matrix formalism for wedge transfer-matrices and boundary and interface condition matrices. This makes it, firstly, very suitable for computational implementations, secondly, very efficient especially in cases with several perfectly bonded homogeneous wedges, and, thirdly, very accurate due to its fully semi-analytic nature. The code developed is tested by solving a large variety of examples, comparing the present results with those obtained by solving closed-form corner-eigenequations deduced by previous authors for specific useful practical configurations, confirming the extremely high accuracy of the present code in the computation of λ and ω. The differences observed in some cases with anisotropic materials are explained by the fact that some of the previous authors did not take the true 3D Coulomb friction law into account.

Power law singularity for cavity collapse in a compressible Euler fluid with Tait–Murnaghan equation of state

Motivated by the high energy focusing found in rapidly collapsing bubbles, which is relevant to implosion processes that concentrate energy density, such as sonoluminescence, we consider a calculation of an empty cavity collapse in a compressible Euler fluid. We review and then use the method based on similarity theory that was previously used to compute the power law exponent n for the collapse of an empty cavity in water during the late stage of the collapse. We extend this calculation by considering different fluids surrounding the cavity, all of which are parametrized by the Tait–Murnaghan equation of state through parameter γ. As a result, we obtain the dependence of n on γ for a wide range of γ, and indeed see that the collapse is sensitive to the equation of state of an outside fluid.

Exact results of dynamical structure factor of Lieb–Liniger model

The dynamical structure factor (DSF) represents a measure of dynamical density–density correlations in a quantum many-body system. Due to the complexity of many-body correlations and quantum fluctuations in a system of an infinitely large Hilbert space, such kind of dynamical correlations often impose a big theoretical challenge. For one-dimensional (1D) quantum many-body systems, qualitative predictions of dynamical response functions are usually carried out by using the Tomonaga–Luttinger liquid (TLL) theory. In this scenario, a precise evaluation of the DSF for a 1D quantum system with arbitrary interaction strength remains a formidable task. In this paper, we use the form factor approach based on algebraic Bethe ansatz theory to calculate precisely the DSF of Lieb–Liniger model with an arbitrary interaction strength at a large scale of particle number. We find that the DSF for a system as large as 2000 particles enables us to depict precisely its line-shape from which the power-law singularity with corresponding exponents in the vicinities of spectral thresholds naturally emerge. It should be noted that, the advantage of our algorithm promises an access to the threshold behavior of dynamical correlation functions, further confirming the validity of nonlinear TLL theory besides Kitanine et al (2012 J. Stat. Mech. P09001). Finally we discuss a comparison of results with the results from the ABACUS method by J-S Caux (2009 J. Math. Phys. 50 095214) as well as from the strongly coupling expansion by Brand and Cherny (2005 Phys. Rev. A 72 033619).

A finite-time singularity in the dynamics of the US equity market: Will the US equity market eventually collapse?

Fitting Dow Jones 30 index data for the 1790–1999 period into a log-periodic power-law singularity (LPPLS) model, the seminal paper by Johansen and Sornette (2001) was the first to show that the US equity growth rate is accelerating such that the market is growing as a power law toward a spontaneous singularity. Their model suggests that the US equity market will reach this critical point in the year 2052 ± 10 years, signaling an abrupt transition to a new regime. This study re-examines this important issue using (i) a novel approach to calibrate the LPPLS model and (ii) a different data set including >20 years of additional data. The extended data account for the dot.com bubble burst (2000), the Global Financial Crisis period (2008–2009), the COVID−19 crisis (2020−2022), and the ongoing Russian–Ukrainian war (starting in 2022), which are all events with severe consequences for the global economy. The calibrated LPPLS model suggests that the US equity market will reach a singularity condition by June 2050.

Observation of magnetic structural universality and jamming transition with NMR

Nuclear magnetic resonance (NMR) has been instrumental in deciphering the structure of proteins. Here we show that transverse NMR relaxation, through its time-dependent relaxation rate, is distinctly sensitive to the structure of complex materials or biological tissues at the mesoscopic scale, from micrometers to tens of micrometers. Based on the ideas of universality, we show analytically and numerically that the time-dependent transverse relaxation rate approaches its long-time limit in a power-law fashion, with the dynamical exponent reflecting the universality class of mesoscopic magnetic structure. The spectral line shape acquires the corresponding non-analytic power law singularity at zero frequency. We experimentally detect the change in the dynamical exponent as a result of the transition into maximally random jammed state characterized by hyperuniform correlations. The relation between relaxational dynamics and magnetic structure opens the way for noninvasive characterization of porous media, complex materials and biological tissues