arXiv subjects

D. Sornette

Publications and source records attributed to D. Sornette.

At least 199 records · Page 11Linked to original sources

arXiv2001

"Slimming" of power law tails by increasing market returns

We introduce a simple generalization of rational bubble models which removes the fundamental problem discovered by [Lux and Sornette, 1999] that the distribution of returns is a power law with exponent less than 1, in contradiction with empirical data. The idea is that the price fluctuations associated with bubbles must on average grow with the mean market return r. When r is larger than the discount rate r_delta, the distribution of returns of the observable price, sum of the bubble component and of the fundamental price, exhibits an intermediate tail with an exponent which can be larger than 1. This regime r>r_delta corresponds to a generalization of the rational bubble model in which the fundamental price is no more given by the discounted value of future dividends. We explain how this is possible. Our model predicts that, the higher is the market remuneration r above the discount rate, the larger is the power law exponent and thus the thinner is the tail of the distribution of price returns.

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arXiv2001

Predictability of catastrophic events: material rupture, earthquakes, turbulence, financial crashes and human birth

We propose that catastrophic events are "outliers" with statistically different properties than the rest of the population and result from mechanisms involving amplifying critical cascades. Applications and the potential for prediction are discussed in relation to the rupture of composite materials, great earthquakes, turbulence and abrupt changes of weather regimes, financial crashes and human parturition (birth).

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arXiv2001

Positive Feedback, Memory and the Predictability of Earthquakes

We review the "critical point" concept for large earthquakes and enlarge it in the framework of so-called "finite-time singularities". The singular behavior associated with accelerated seismic release is shown to result from a positive feedback of the seismic activity on its release rate. The most important mechanisms for such positive feedback are presented. We introduce and solve analytically a novel simple model of geometrical positive feedback in which the stress shadow cast by the last large earthquake is progressively fragmented by the increasing tectonic stress. Finally, we present a somewhat speculative figure that tends to support a mechanism based on the decay of stress shadows. This figure suggests that a large earthquake in Southern California of size similar to the 1812 great event is maturing.

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arXiv2001

Significance of log-periodic precursors to financial crashes

We clarify the status of log-periodicity associated with speculative bubbles preceding financial crashes. In particular, we address Feigenbaum's [2001] criticism and show how it can be rebuked. Feigenbaum's main result is as follows: ``the hypothesis that the log-periodic component is present in the data cannot be rejected at the 95% confidence level when using all the data prior to the 1987 crash; however, it can be rejected by removing the last year of data.'' (e.g., by removing 15% of the data closest to the critical point). We stress that it is naive to analyze a critical point phenomenon, i.e., a power law divergence, reliably by removing the most important part of the data closest to the critical point. We also present the history of log-periodicity in the present context explaining its essential features and why it may be important. We offer an extension of the rational expectation bubble model for general and arbitrary risk-aversion within the general stochastic discount factor theory. We suggest guidelines for using log-periodicity and explain how to develop and interpret statistical tests of log-periodicity. We discuss the issue of prediction based on our results and the evidence of outliers in the distribution of drawdowns. New statistical tests demonstrate that the 1% to 10% quantile of the largest events of the population of drawdowns of the Nasdaq composite index and of the Dow Jones Industrial Average index belong to a distribution significantly different from the rest of the population. This suggests that very large drawdowns result from an amplification mechanism that may make them more predictable than smaller market moves.

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arXiv2001

Oscillatory Finite-Time Singularities in Finance, Population and Rupture

We present a simple two-dimensional dynamical system where two nonlinear terms, exerting respectively positive feedback and reversal, compete to create a singularity in finite time decorated by accelerating oscillations. The power law singularity results from the increasing growth rate. The oscillations result from the restoring mechanism. As a function of the order of the nonlinearity of the growth rate and of the restoring term, a rich variety of behavior is documented analytically and numerically. The dynamical behavior is traced back fundamentally to the self-similar spiral structure of trajectories in phase space unfolding around an unstable spiral point at the origin. The interplay between the restoring mechanism and the nonlinear growth rate leads to approximately log-periodic oscillations with remarkable scaling properties. Three domains of applications are discussed: (1) the stock market with a competition between nonlinear trend-followers and nonlinear value investors; (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity; (3) the failure of a material subjected to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing.

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arXiv2001

Theory of self-similar oscillatory finite-time singularities in Finance, Population and Rupture

This is a short letter summarizing the long paper cond-mat/0106047 in which we present a simple two-dimensional dynamical system reaching a singularity in finite time decorated by accelerating oscillations due to the interplay between nonlinear positive feedback and reversal in the inertia. This provides a fundamental equation for the dynamics of (1) stock market prices in the presence of nonlinear trend-followers and nonlinear value investors, (2) the world human population with a competition between a population-dependent growth rate and a nonlinear dependence on a finite carrying capacity and (3) the failure of a material subject to a time-varying stress with a competition between positive geometrical feedback on the damage variable and nonlinear healing. The rich fractal scaling properties of the dynamics are traced back to the self-similar spiral structure in phase space unfolding around an unstable spiral point at the origin.

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arXiv2001

The Kalman-Levy filter

The Kalman filter combines forecasts and new observations to obtain an estimation which is optimal in the sense of a minimum average quadratic error. The Kalman filter has two main restrictions: (i) the dynamical system is assumed linear and (ii) forecasting errors and observational noises are taken Gaussian. Here, we offer an important generalization to the case where errors and noises have heavy tail distributions such as power laws and Lévy laws. The main tool needed to solve this ``Kalman-Lévy'' filter is the ``tail-covariance'' matrix which generalizes the covariance matrix in the case where it is mathematically ill-defined (i.e. for power law tail exponents $μ\leq 2$). We present the general solution and discuss its properties on pedagogical examples. The standard Kalman-Gaussian filter is recovered for the case $μ= 2$. The optimal Kalman-Lévy filter is found to deviate substantially fro the standard Kalman-Gaussian filter as $μ$ deviates from 2. As $μ$ decreases, novel observations are assimilated with less and less weight as a small exponent $μ$ implies large errors with significant probabilities. In terms of implementation, the price-to-pay associated with the presence of heavy tail noise distributions is that the standard linear formalism valid for the Gaussian case is transformed into a nonlinear matrice equation for the Kalman-Lévy filter. Direct numerical experiments in the univariate case confirms our theoretical predictions.

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arXiv2001

General framework for a portfolio theory with non-Gaussian risks and non-linear correlations

Using a family of modified Weibull distributions, encompassing both sub-exponentials and super-exponentials, to parameterize the marginal distributions of asset returns and their natural multivariate generalizations, we give exact formulas for the tails and for the moments and cumulants of the distribution of returns of a portfolio make of arbitrary compositions of these assets. Using combinatorial and hypergeometric functions, we are in particular able to extend previous results to the case where the exponents of the Weibull distributions are different from asset to asset and in the presence of dependence between assets. We treat in details the problem of risk minimization using two different measures of risks (cumulants and value-at-risk) for a portfolio made of two assets and compare the theoretical predictions with direct empirical data. While good agreement is found, the remaining discrepancy between theory and data stems from the deviations from the Weibull parameterization for small returns. Our extended formulas enable us to determine analytically the conditions under which it is possible to ``have your cake and eat it too'', i.e., to construct a portfolio with both larger return and smaller ``large risks''.

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arXiv2001

From Rational Bubbles to Crashes

We study and generalize in various ways the model of rational expectation (RE) bubbles introduced by Blanchard and Watson in the economic literature. First, bubbles are argued to be the equivalent of Goldstone modes of the fundamental rational pricing equation, associated with the symmetry-breaking introduced by non-vanishing dividends. Generalizing bubbles in terms of multiplicative stochastic maps, we summarize the result of Lux and Sornette that the no-arbitrage condition imposes that the tail of the return distribution is hyperbolic with an exponent mu<1. We then extend the RE bubble model to arbitrary dimensions d and, with the renewal theory for products of random matrices applied to stochastic recurrence equations, we extend the theorem of Lux and Sornette to demonstrate that the tails of the unconditional distributions follow power laws, with the same asymptotic tail exponent mu<1 for all assets. Two extensions (the crash hazard rate model and the non-stationary growth rate model) of the RE bubble model provide ways of reconciliation with the stylized facts of financial data. The later model allows for an understanding of the breakdown of the fundamental valuation formula as deeply associated with a spontaneous breaking of the price symmetry. Its implementation for multi-dimensional bubbles explains why the tail index mu seems to be the same for any group af assets as observed empirically. This work begs for the introduction of a generalized field theory which would be able to capture the spontaneous breaking of symmetry, recover the fundamental valuation formula in the normal economic case and extend it to the still unexplored regime where the economic growth rate is larger than the discount growth rate.

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arXiv2001

Multi-dimensional Rational Bubbles and fat tails: application of stochastic regression equations to financial speculation

We extend the model of rational bubbles of Blanchard and of Blanchard and Watson to arbitrary dimensions d: a number d of market time series are made linearly interdependent via d times d stochastic coupling coefficients. We first show that the no-arbitrage condition imposes that the non-diagonal impacts of any asset i on any other asset j different from i has to vanish on average, i.e., must exhibit random alternative regimes of reinforcement and contrarian feedbacks. In contrast, the diagonal terms must be positive and equal on average to the inverse of the discount factor. Applying the results of renewal theory for products of random matrices to stochastic recurrence equations (SRE), we extend the theorem of Lux and Sornette (cond-mat/9910141) and demonstrate that the tails of the unconditional distributions associated with such d-dimensional bubble processes follow power laws (i.e., exhibit hyperbolic decline), with the same asymptotic tail exponent mu<1 for all assets. The distribution of price differences and of returns is dominated by the same power-law over an extended range of large returns. This small value mu<1 of the tail exponent has far-reaching consequences in the non-existence of the means and variances. Although power-law tails are a pervasive feature of empirical data, the numerical value mu<1 is in disagreement with the usual empirical estimates mu approximately equal to 3. It, therefore, appears that generalizing the model of rational bubbles to arbitrary dimensions does not allow us to reconcile the model with these stylized facts of financial data. The non-stationary growth rational bubble model seems at present the only viable solution (see cond-mat/0010112).

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arXiv2000

Solitary waves and supersonic reaction front in metastable solids

Motivated by an increasing number of remarkable experimental observations on the role of pressure and shear stress in solid reactions,explosions and detonations,we present a simple toy model that embodies nonlinear elasticity and dispersion as well as chemical or phase transformation.This generalization of the Toda Lattice provides an effective model for the description of the organization during an abrupt transformation in a solid.One of the challenges is to capture both the equilibrium degrees of freedom as well as to quantify the possible role of out-of-equilibrium perturbations.In the Toda Lattice,we verify that the particle velocities converge in distribution towards the Maxwell-Boltzmann distribution,thus allowing us to define a bona-fide temperature.In addition,the balance between nonlinearity and wave dispersion may create solitary waves that act as energy traps.In the presence of reactive chemistry,we show that the trapping of the released chemical energy in solitary waves that are excited by an initial perturbation provides a positive feedback that enhances the reaction rate and leads to supersonic explosion front propagation.These modes of rupture observed in our model may provide a first-order description of ultrafast reactions of heterogeneous mixtures under mechanical loading.

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arXiv2000

Statistical Asynchronous Regression: Determining the Relationship Between two Quantities that are not Measured Simultaneously

We introduce the Statistical Asynchronous Regression (SAR) method: a technique for determining a relationship between two time varying quantities without simultaneous measurements of both quantities. We require that there is a time invariant, monotonic function Y = u(X) relating the two quantities, Y and X. In order to determine u(X), we only need to know the statistical distributions of X and Y. We show that u(X) is the change of variables that converts the distribution of X into the distribution of Y, while conserving probability. We describe an algorithm for implementing this method and apply it to several example distributions. We also demonstrate how the method can separate spatial and temporal variations from a time series of energetic electron flux measurements made by a spacecraft in geosynchronous orbit. We expect this method will be useful to the general problem of spacecraft instrument calibration. We also suggest some applications of the SAR method outside of space physics.

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arXiv2000

Functional Renormalization Prediction of Rupture

We develop theoretical formulas for the prediction of the rupture of systems which are known to exhibit a critical behavior, based solely on the knowledge of the early time evolution of an observable, such as the acoustic emission rate as a function of time or of stress. From the parameterization of such early time evolution in terms of a low-order polynomial, we use the functional renormalization approach introduced by Yukalov and Gluzman to transform this polynomial into a function which is asymptotically a power law. The value of the critical time tc,conditioned on the assumption that tc exists, is thus determined from the knowledge of the coefficients of the polynomials. We test with success this prediction scheme with respect to the order of the polynomials and as a function of noise.

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arXiv2000

Characterization of the frequency of extreme events by the Generalized Pareto Distribution

Based on recent results in extreme value theory, we use a new technique for the statistical estimation of distribution tails. Specifically, we use the Gnedenko-Pickands-Balkema-de Haan theorem, which gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD). Useful in finance, insurance, hydrology, we investigate here the earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law and analyze shallow earthquakes (depth h < 70 km) in the Harvard catalog over the period 1977-2000 in 18 seismic zones. The whole GPD is found to approximate the tails of the seismic moment distributions quite well above moment-magnitudes larger than mW=5.3 and no statistically significant regional difference is found for subduction and transform seismic zones. We confirm that the b-value is very different in mid-ocean ridges compared to other zones (b=1.50=B10.09 versus b=1.00=B10.05 corresponding to a power law exponent close to 1 versus 2/3) with a very high statistical confidence. We propose a physical mechanism for this, contrasting slow healing ruptures in mid-ocean ridges with fast healing ruptures in other zones. Deviations from the GPD at the very end of the tail are detected in the sample containing earthquakes from all major subduction zones (sample size of 4985 events). We propose a new statistical test of significance of such deviations based on the bootstrap method. The number of events deviating from the tails of GPD in the studied data sets (15-20 at most) is not sufficient for determining the functional form of those deviations. Thus, it is practically impossible to give preference to one of the previously suggested parametric families describing the ends of tails of seismic moment distributions.

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arXiv2000

Fokker-Planck equation of distributions of financial returns and power laws

Our purpose is to relate the Fokker-Planck formalism proposed by [Friedrich et al., Phys. Rev. Lett. 84, 5224 (2000)] for the distribution of stock market returns to the empirically well-established power law distribution with an exponent in the range 3-5. We show how to use Friedrich et al.'s formalism to predict that the distribution of returns is indeed asymptotically a power law with an exponent mu that can be determined from the Kramers-Moyal coefficients determined by Friedrich et al. However, with their values determined for the U.S. dollar-German mark exchange rates, the exponent mu predicted from their theory is found around 12, in disagreement with the often-quoted value between 3 and 5. This could be explained by the fact that the large asymptotic value of 12 does not apply to real data that lie still far from the stationary state of the Fokker-Planck description. Another possibility is that power laws are inadequate. The mechanism for the power law is based on the presence of multiplicative noise across time-scales, which is different from the multiplicative noise at fixed time-scales implicit in the ARCH models developed in the Finance literature.

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arXiv2000

Multifractal returns and Hierarchical Portfolio Theory

We extend and test empirically the multifractal model of asset returns based on a multiplicative cascade of volatilities from large to small time scales. The multifractal description of asset fluctuations is generalized into a multivariate framework to account simultaneously for correlations across times scales and between a basket of assets. The reported empirical results show that this extension is pertinent for financial modelling. The second part of the paper applies this theory to portfolio optimisation. Our multi-scale description allows us to characterize the portfolio return distribution at all time scales simultaneously. The portfolio composition is predicted to change with the investment time horizon (i.e., the time scale) in a way that can be fully determined once an adequate measure of risk is chosen. We discuss the use of the fourth-order cumulant and of utility functions. While the portfolio volatility can be optimized in some cases for all time horizons, the kurtosis and higher normalized cumulants cannot be simultaneously optimized. For a fixed investment horizon, we study in details the influence of the number of periods, i.e., of the number of rebalancing of the portfolio. For the large risks quantified by the cumulants of order larger than two, the number of periods has a non-trivial influence, in contrast with Tobin's result valid in the mean-variance framework. This theory provides a fundamental framework for the conflicting optimization involved in the different time horizons and quantifies systematically the trade-offs for an optimal inter-temporal portfolio optimization.

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arXiv2000

Self-Consistent Theory of Rupture by Progressive Diffuse Damage

We analyze a self-consistent theory of crack growth controlled by a cumulative damage variable d(t) dependent on stress history. As a function of the damage exponent $m$, which controls the rate of damage dd/dt \propto sigma^m as a function of local stress $σ$, we find two regimes. For 0 < m < 2, the model predicts a finite-time singularity. This retrieves previous results by Zobnin for m=1 and by Bradley and Wu for 0 < m < 2. To improve on this self-consistent theory which neglects the dependence of stress on damage, we apply the functional renormalization method of Yukalov and Gluzman and find that divergences are replaced by singularities with exponents in agreement with those found in acoustic emission experiments. For m =2 and m > 2, the rupture dynamics is not defined without the introduction of a regularizing scheme. We investigate three regularization schemes involving respectively a saturation of damage, a minimum distance of approach to the crack tip and a fixed stress maximum. In the first and third schemes, the finite-time singularity is replaced by a crack dynamics defined for all times but which is controlled by either the existence of a microscopic scale at which the stress is regularized or by the maximum sustainable stress. In the second scheme, a finite-time singularity is again found. In the first two schemes within this regime m > 2, the theory has no continuous limit.

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arXiv2000

Artifactual log-periodicity in finite size data: Relevance for earthquake aftershocks

The recently proposed discrete scale invariance and its associated log-periodicity are an elaboration of the concept of scale invariance in which the system is scale invariant only under powers of specific values of the magnification factor. We report on the discovery of a novel mechanism for such log-periodicity relying solely on the manipulation of data. This ``synthetic'' scenario for log-periodicity relies on two steps: (1) the fact that approximately logarithmic sampling in time corresponds to uniform sampling in the logarithm of time; and (2) a low-pass-filtering step, as occurs in constructing cumulative functions, in maximum likelihood estimations, and in de-trending, reddens the noise and, in a finite sample, creates a maximum in the spectrum leading to a most probable frequency in the logarithm of time. We explore in detail this mechanism and present extensive numerical simulations. We use this insight to analyze the 27 best aftershock sequences studied by Kisslinger and Jones [1991] to search for traces of genuine log-periodic corrections to Omori's law, which states that the earthquake rate decays approximately as the inverse of the time since the last main shock. The observed log-periodicity is shown to almost entirely result from the ``synthetic scenario'' owing to the data analysis. From a statistical point of view, resolving the issue of the possible existence of log-periodicity in aftershocks will be very difficult as Omori's law describes a point process with a uniform sampling in the logarithm of the time. By construction, strong log-periodic fluctuations are thus created by this logarithmic sampling.

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