arXiv subjects

D. Sornette

Publications and source records attributed to D. Sornette.

At least 235 records · Page 13Linked to original sources

arXiv1999

"Nonlinear" covariance matrix and portfolio theory for non-Gaussian multivariate distributions

This paper offers a precise analytical characterization of the distribution of returns for a portfolio constituted of assets whose returns are described by an arbitrary joint multivariate distribution. In this goal, we introduce a non-linear transformation that maps the returns onto gaussian variables whose covariance matrix provides a new measure of dependence between the non-normal returns, generalizing the covariance matrix into a non-linear fractional covariance matrix. This nonlinear covariance matrix is chiseled to the specific fat tail structure of the underlying marginal distributions, thus ensuring stability and good-conditionning. The portfolio distribution is obtained as the solution of a mapping to a so-called phi-q field theory in particle physics, of which we offer an extensive treatment using Feynman diagrammatic techniques and large deviation theory, that we illustrate in details for multivariate Weibull distributions. The main result of our theory is that minimizing the portfolio variance (i.e. the relatively ``small'' risks) may often increase the large risks, as measured by higher normalized cumulants. Extensive empirical tests are presented on the foreign exchange market that validate satisfactorily the theory. For ``fat tail'' distributions, we show that an adequete prediction of the risks of a portfolio relies much more on the correct description of the tail structure rather than on their correlations.

cond-mat.stat-mech↗

arXiv1999

Financial ``Anti-Bubbles'': Log-Periodicity in Gold and Nikkei collapses

We propose that imitation between traders and their herding behaviour not only lead to speculative bubbles with accelerating over-valuations of financial markets possibly followed by crashes, but also to ``anti-bubbles'' with decelerating market devaluations following all-time highs. For this, we propose a simple market dynamics model in which the demand decreases slowly with barriers that progressively quench in, leading to a power law decay of the market price decorated by decelerating log-periodic oscillations. We document this behaviour on the Japanese Nikkei stock index from 1990 to present and on the Gold future prices after 1980, both after their all-time highs. We perform simultaneously a parametric and non-parametric analysis that are fully consistent with each other. We extend the parametric approach to the next order of perturbation, comparing the log-periodic fits with one, two and three log-frequencies, the latter one providing a prediction for the general trend in the coming years. The non-parametric power spectrum analysis shows the existence of log-periodicity with high statistical significance, with a prefered scale ratio of $λ\approx 3.5$ for the Nikkei index $λ\approx 1.9$ for the Gold future prices, comparable to the values obtained for speculative bubbles leading to crashes.

cond-mat.stat-mech↗

arXiv1998

Minimizing volatility increases large risks

We introduce a faithful representation of the heavy tail multivariate distribution of asset returns, as parsimonous as the Gaussian framework. Using calculation techniques of functional integration and Feynman diagrams borrowed from particle physics, we characterize precisely, through its cumulants of high order, the distribution of wealth variations of a portfolio composed of an arbitrary mixture of assets. The portfolio which minimizes the variance, i.e. the relatively small risks, often increases larger risks as measured by higher normalized cumulants and by the Value-at-risk.

cond-mat.stat-mech↗

arXiv1998

Data-Adaptive Wavelets and Multi-Scale Singular Spectrum Analysis

Using multi-scale ideas from wavelet analysis, we extend singular-spectrum analysis (SSA) to the study of nonstationary time series of length $N$ whose intermittency can give rise to the divergence of their variance. SSA relies on the construction of the lag-covariance matrix C on M lagged copies of the time series over a fixed window width W to detect the regular part of the variability in that window in terms of the minimal number of oscillatory components; here W = M Dt, with Dt the time step. The proposed multi-scale SSA is a local SSA analysis within a moving window of width M <= W <= N. Multi-scale SSA varies W, while keeping a fixed W/M ratio, and uses the eigenvectors of the corresponding lag-covariance matrix C_M as a data-adaptive wavelets; successive eigenvectors of C_M correspond approximately to successive derivatives of the first mother wavelet in standard wavelet analysis. Multi-scale SSA thus solves objectively the delicate problem of optimizing the analyzing wavelet in the time-frequency domain, by a suitable localization of the signal's covariance matrix. We present several examples of application to synthetic signals with fractal or power-law behavior which mimic selected features of certain climatic and geophysical time series. A real application is to the Southern Oscillation index (SOI) monthly values for 1933-1996. Our methodology highlights an abrupt periodicity shift in the SOI near 1960. This abrupt shift between 4 and 3 years supports the Devil's staircase scenario for the El Nino/Southern Oscillation phenomenon.

chao-dyn↗

arXiv1998

Testing self-organized criticality by induced seismicity

We examine the hypothesis proposed in recent years by several authors that the crust is in a self-organized critical (SOC) state by exploring how the SOC concept can help in understanding the observed earthquake clustering on relatively narrow fault domains and the phenomenon of induced seismicity. We review the major reported cases of induced seismicity in various parts of the world and find that both pore pressure changes (+/-p) and mass transfers (+/-m) leading to incremental deviatoric stresses of <<1 MPa are sufficient to trigger seismic instabilities in the uppermost crust with magnitude ranging up to 7.0 in otherwise historically aseismic areas. Once triggered, stress variations of at least 1 order of magnitude less but still larger than the ~0.01 MPa tidal stress are enough to sustain seismic activity. We argue that these observations are in accord with the SOC hypothesis as they show that a significant fraction of the crust is not far from instability and can thus be made unstable by minute perturbations. Not all perturbations, however, trigger seismic activity; this is also compatible with the SOC hypothesis which embodies naturally the existence of large heterogeneities in the stress field. In conclusion, the properties of induced seismicity and their rationalization in terms of the SOC concept provide further evidence that potential seismic hazards extend over a much larger area than that where earthquakes are frequent.

cond-mat.stat-mech↗

arXiv1998

Solid friction at high sliding velocities: an explicit 3D dynamical SPH approach

We present realistic 3D numerical simulations of elastic bodies sliding on top of each other in a regime of velocities ranging from meters to tens of meters per second using the so-called Smoothed Particle Hydrodynamics (SPH) method. Our investigations are restricted to regimes of pressure and roughness where only elastic deformations occur between asperities at the contact surface between the slider block and the substrate. In this regime, solid friction is due to the generation of vibrational radiations which are subsequently damped out. We study periodic commensurate and incommensurate asperities and various types of disordered surfaces. We report the evidence of a transition from zero (or non-measurable $μ< 0.001$) friction to a finite friction as the normal pressure increases above about $10^6~Pa$. For larger normal pressures (up to $10^9~Pa$), we find a remarkably universal value for the friction coefficient $μ\approx 0.06$, which is independent of the internal dissipation strength over three order of magnitudes, and independent of the detailled nature of the slider block-substrate interactions. We find that disorder may either decrease or increase $μ$ due to the competition between two effects: disorder detunes the coherent vibrations of the asperties that occur in the periodic case, leading to weaker acoustic radiation and thus weaker damping. On the other hand, large disorder leads to stronger vibration amplitudes at local asperities and thus stronger damping. Our simulations have confirmed the existence of jumps over steps or asperities of the slider blocks occurring at the largest velocities studied ($10~m/s$). These jumps lead to chaotic motions similar to the bouncing-ball problem. We find a velocity strengthening with a doubling of the friction coefficient as the velocity increases from $1~m/s$ to $10~m/s$.

cond-mat.stat-mech↗

arXiv1998

Mechanochemistry: an hypothesis for shallow earthquakes

We advance the novel hypothesis that water in the presence of finite localized strain within fault gouges may lead to the phase transformation of stable minerals into metastable polymorphs of higher free energy density. Under increasing strain, the transformed minerals eventually become unstable, as shown from an application of Landau theory of structural phase transitions. We propose that this instability leads to an explosive transformation, creating a slightly supersonic shock wave propagating along the altered fault core leaving a wake of shaking fragments. As long as the resulting high-frequency acoustic waves remain of sufficient amplitude to lead to a fluidization of the fault core, the fault is unlocked and free to slip under the effect of the tectonic stress, thus releasing the elastic part of the stored energy. We briefly discuss observations that could be understood within this theory.

cond-mat.mtrl-sci↗

arXiv1998

Earthquakes: from chemical alteration to mechanical rupture

In the standard rebound theory of earthquakes, elastic deformation energy is progressively stored in the crust until a threshold is reached at which it is suddenly released in an earthquake. We review three important paradoxes, the strain paradox, the stress paradox and the heat flow paradox, that are difficult to account for in this picture, either individually or when taken together. Resolutions of these paradoxes usually call for additional assumptions on the nature of the rupture process (such as novel modes of deformations and ruptures) prior to and/or during an earthquake, on the nature of the fault and on the effect of trapped fluids within the crust at seismogenic depths. We review the evidence for the essential importance of water and its interaction with the modes of deformations. Water is usually seen to have mainly the mechanical effect of decreasing the normal lithostatic stress in the fault core on one hand and to weaken rock materials via hydrolytic weakening and stress corrosion on the other hand. We also review the evidences that water plays a major role in the alteration of minerals subjected to finite strains into other structures in out-of-equilibrium conditions. This suggests novel exciting routes to understand what is an earthquake, that requires to develop a truly multidisciplinary approach involving mineral chemistry, geology, rupture mechanics and statistical physics.

cond-mat.dis-nn↗

arXiv1998

Gauge theory of Finance?

Some problems with the recent stimulating proposal of a ``Gauge Theory of Finance'' by Ilinski and collaborators are outlined. First, the derivation of the log-normal distribution is shown equivalent both in information and mathematical content to the simpler and well-known derivation, dating back from Bachelier and Samuelson. Similarly, the re-derivation of Black-Scholes equation is shown equivalent to the standard one because the limit of no uncertainty is equivalent to the standard risk-free replication argument. Both re-derivations of the log-normality and Black-Scholes result do not provide a test of the theory because it is degenerate in the limits where these results apply. Third, the choice of the exponential form a la Boltzmann, of the weight of a given market configuration, is a key postulate that requires justification. In addition, the ``Gauge Theory of Finance'' seems to lead to ``virtual'' arbitrage opportunities for pure Markov random walk market when there should be none. These remarks are offered in the hope to improve the formulation of the ``Gauge Theory of Finance'' into a coherent and useful framework.

cond-mat.stat-mech↗

arXiv1998

An Observational Test of the Critical Earthquake Concept

We test the concept that seismicity prior to a large earthquake can be understood in terms of the statistical physics of a critical phase transition. In this model, the cumulative seismic strain release increases as a power-law time-to-failure before the final event. Furthermore, the region of correlated seismicity predicted by this model is much greater than would be predicted from simple elasto-dynamic interactions. We present a systematic procedure to test for the accelerating seismicity predicted by the critical point model and to identify the region approaching criticality, based on a comparison between the observed cumulative energy (Benioff strain) release and the power-law behavior predicted by theory. This method is used to find the critical region before all earthquakes along the San Andreas system since 1950 with M 6.5. The statistical significance of our results is assessed by performing the same procedure on a large number of randomly generated synthetic catalogs. The null hypothesis, that the observed acceleration in all these earthquakes could result from spurious patterns generated by our procedure in purely random catalogs, is rejected with 99.5% confidence. An empirical relation between the logarithm of the critical region radius (R) and the magnitude of the final event (M) is found, such that log R μ0.5 M, suggesting that the largest probable event in a given region scales with the size of the regional fault network.

cond-mat.stat-mech↗

arXiv1998

Evidence of discrete scale invariance in DLA and time-to-failure by canonical averaging

Discrete scale invariance, which corresponds to a partial breaking of the scaling symmetry, is reflected in the existence of a hierarchy of characteristic scales l0, c l0, c^2 l0,... where c is a preferred scaling ratio and l0 a microscopic cut-off. Signatures of discrete scale invariance have recently been found in a variety of systems ranging from rupture, earthquakes, Laplacian growth phenomena, ``animals'' in percolation to financial market crashes. We believe it to be a quite general, albeit subtle phenomenon. Indeed, the practical problem in uncovering an underlying discrete scale invariance is that standard ensemble averaging procedures destroy it as if it was pure noise. This is due to the fact, that while c only depends on the underlying physics, l0 on the contrary is realisation-dependent. Here, we adapt and implement a novel so-called ``canonical'' averaging scheme which re-sets the l0 of different realizations to approximately the same value. The method is based on the determination of a realization-dependent effective critical point obtained from, e.g., a maximum susceptibility criterion. We demonstrate the method on diffusion limited aggregation and a model of rupture.

cond-mat.stat-mech↗

arXiv1998

The sharp peak-flat trough pattern and critical speculation

We find empirically a characteristic sharp peak-flat trough pattern in a large set of commodity prices. We argue that the sharp peak structure reflects an endogenous inter-market organization, and that peaks may be seen as local ``singularities'' resulting from imitation and herding. These findings impose a novel stringent constraint on the construction of models. Intermittent amplification is not sufficient and nonlinear effects seem necessary to account for the observations.

cond-mat.stat-mech↗

arXiv1998

Multiplicative processes and power laws

[Takayasu et al., Phys. Rev.Lett. 79, 966 (1997)] revisited the question of stochastic processes with multiplicative noise, which have been studied in several different contexts over the past decades. We focus on the regime, found for a generic set of control parameters, in which stochastic processes with multiplicative noise produce intermittency of a special kind, characterized by a power law probability density distribution. We briefly explain the physical mechanism leading to a power law pdf and provide a list of references for these results dating back from a quarter of century. We explain how the formulation in terms of the characteristic function developed by Takayasu et al. can be extended to exponents $μ>2$, which explains the ``reason of the lucky coincidence''. The multidimensional generalization of (\ref{eq1}) and the available results are briefly summarized. The discovery of stretched exponential tails in the presence of the cut-off introduced in \cite{Taka} is explained theoretically. We end by briefly listing applications.

cond-mat.stat-mech↗

arXiv1998

``String'' formulation of the Dynamics of the Forward Interest Rate Curve

We propose a formulation of the term structure of interest rates in which the forward curve is seen as the deformation of a string. We derive the general condition that the partial differential equations governing the motion of such string must obey in order to account for the condition of absence of arbitrage opportunities. This condition takes a form similar to a fluctuation-dissipation theorem, albeit on the same quantity (the forward rate), linking the bias to the covariance of variation fluctuations. We provide the general structure of the models that obey this constraint in the framework of stochastic partial (possibly non-linear) differential equations. We derive the general solution for the pricing and hedging of interest rate derivatives within this framework, albeit for the linear case (we also provide in the appendix a simple and intuitive derivation of the standard European option problem). We also show how the ``string'' formulation simplifies into a standard N-factor model under a Galerkin approximation.

cond-mat.stat-mech↗

arXiv1998

Discrete scale invariance in turbulence?

Based on theoretical argument and experimental evidence, we conjecture that structure functions of turbulent times series exhibit log-periodic modulations decorating their power law dependence. In order to provide ironclad experimental evidence, we stress the need for novel methods of averaging and propose a novel ``canonical'' averaging scheme for the analysis of structure factors of turbulent flows. The strategy is to determine the scale $r_c$ at which the dissipation rate is the largest in a given turn-over time series. This specific scale $r_c$ translates into a specific ``phase'' in the logarithm of the scale which, when used as the origin, allows one to phase up the different measurements of a structure factor $S_p(r) = A_p (\barεr)^{p/3}$ in different turn-over time realizations. We expect, as in Laplacian growth and in rupture, that the log-periodic oscillations will be reinforced by this canonical averaging. Demonstrating unambiguously the presence of log-periodicity and thus of discrete scale invariance (DSI) in turbulent time-series would provide an important step towards a direct demonstration of the Kolmogorov cascade or at least of its hierarchical imprint.

cond-mat.stat-mech↗

arXiv1998

The Dynamics of the Forward Interest Rate Curve with Stochastic String Shocks

This paper offers a new class of models of the term structure of interest rates. We allow each instantaneous forward rate to be driven by a different stochastic shock, constrained in such a way as to keep the forward rate curve continuous. We term the process followed by the shocks to the forward curve ``stochastic strings'', and construct them as the solution to stochastic partial differential equations, that allow us to offer a variety of interesting parametrizations. The models can produce, with parsimony, any sort of correlation pattern among forward rates of different maturities. This feature makes the models consistent with any panel dataset of bond prices, not requiring the addition of error terms in econometric models. Interest rate options can easily be priced by simulation. However, options can only be perfectly hedged by trading in bonds of all maturities available.

cond-mat.stat-mech↗

arXiv1998

Stretched exponential distributions in Nature and Economy: ``Fat tails'' with characteristic scales

To account quantitatively for many reported ``natural'' fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok temperature variations, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also briefly discuss its potential for the distribution of earthquake sizes and fault displacements and earth temperature variations over the last 400 000 years. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions.

cond-mat.stat-mech↗

arXiv1997

Conditions for abrupt failure in the democratic fiber bundle model

We argue that the existence of abrupt failure in the democratic fiber bundle model is more general than concluded by da Silveira in his comment (cond-mat/9709327). We refute his claim that the nature of the rupture process in the DFBM depends on the ``disorder distribution only via its large failure strength behavior''.

cond-mat.stat-mech↗