Stochastic volatility of financial markets as the fluctuating rate of trading: An empirical study

We present an empirical study of the subordination hypothesis for a stochastic time series of a stock price. The fluctuating rate of trading is identified with the stochastic variance of the stock price, as in the continuous-time random walk (CTRW) framework. The probability distribution of the stock price changes (log-returns) for a given number of trades N is found to be approximately Gaussian. The probability distribution of N for a given time interval Δt is non-Poissonian and has an exponential tail for large N and a sharp cutoff for small N. Combining these two distributions produces a non-trivial distribution of log-returns for a given time interval Δt, which has exponential tails and a Gaussian central part, in agreement with empirical observations.     

Das Fluktuationsspektrum von Elektronen in einem stationären Nichtgleichgewichtszustand

The Boltzmann equation is used to calculate the time correlation function and the fluctuation spectrum for electrons obeying classical statistics. The stationary joint distribution for one electron to be initially ink ₀=k(0) and finally ink=k(t) is given by the product of the conditional probability and the stationary distribution. These quantities can be found from the Boltzmann equation if there exists, for any initial distribution, a unique solution which satisfies the Markov equation and tends to a stationary solution for large times under stationary conditions. It is proved that these conditions hold for linear collision operators and in the relaxation approximation. General operator expressions for the fluctuation spectrum and the differential conductivity in a stationary electric field are given, which can be evaluated within the usual approximation schemes known for the stationary, nonequilibrium solutions of the Boltzmann equation. In equilibrium they reproduce the classical fluctuation dissipation theorem. In a nonequilibrium state they define a noise temperature depending on the field. In the relaxation approximation and for polynomial band structure the exact solution can be found. For parabolic and biparabolic spherical bands the result is discussed explicitly.     

Block error rate of FSO links over non-Kolmogorov turbulence with exponentiated Weibull distribution

In this paper, we model block error rate (BLER) performance of optical wireless communication systems in strong turbulence slant channels with very slow fading and additive Gaussian noise, using the Exponentiated Weibull distribution models. In a communication system, which transmits data in blocks of N bits, the BLER probability P(M, N) of more than M bit errors in a block is particularly useful in evaluating the wireless optical channel performance. The joint effects of the beam wander and spread, pointing errors and the spectral index of non-Kolmogorov turbulence on system's performance are included. The obtained results can be essential for the designing of such links under real circumstances.     

Relating structure and flow of soft colloids

To relate the complex macroscopic flow of soft colloids to details of its microscopic equilibrium and non-equilibrium structure is still one big challenge in soft matter science. We investigated several well-defined colloidal model systems like star polymers or diblock copolymer micelles by linear/non-linear rheology, static/dynamic light scattering (SLS/DLS) and small angle neutron scattering (SANS). In addition, in-situ SANS experiments during shear (Rheo-SANS) revealed directly shear induced structural changes on a microscopic level. Varying the molecular architecture of the individual colloidal particle as well as particle-particle interactions and covering at the same time a broad concentration range from the very dilute to highly concentrated, glassy regime, we could separate contributions from intra- and inter-particle softness. Both can be precisely “tuned” by varying systematically the functionality, 6 ≤ f≤ 64, for star polymers or aggregation number, 30 ≤ N _agg ≤ 1000 for diblock copolymer micelles, as well as the degree of polymerization of the individual polymer arm 100 ≤ D _p ≤ 3000. In dilute solutions, the characteristic shear rate at which deformation of the soft colloid is observed can be related to the Zimm time of the polymeric corona. In concentrated solutions, we validated a generalized Stokes-Einstein approach to describe the increase in macroscopic viscosity and mesoscopic self diffusion coefficient on approaching the glassy regime. Both can be explained in terms of an ultra-soft interaction potential. Moreover, non-equilibrium structure factors are obtained by Rheo-SANS. All experimental results are in excellent quantitative agreement with recent theoretical predictions.     

Asymptotic dynamics,non-critical and critical fluctuations for a geometric long-range interacting model

We study the dynamics of geometric spin system on the torus with long-range interaction. As the number of particles goes to infinity, the process converges to a deterministic, dynamical magnetization field that satisfies an Euler equation (law of large numbers). Its stable steady states are related to the limits of the equilibrium measures (Gibbs states) of the finite particle system. A related equation holds for the magnetization densities, for which the property of propagation of chaos also is established. We prove a dynamical central limit theorem with an infinite-dimensional Ornstein-Uhlenbeck process as a limiting fluctuation process. At the critical temperature of a ferromagnetic phase transition, both a tighter quantity scaling and a time scaling is required to obtain convergence to a one-dimensional critical fluctuation process with constant magnetization fields, which has a non-Gaussian invariant distribution. Similarly, at the phase transition to an antiferromagnetic state with frequencyp ₀, the fluctuation process with critical scaling converges to a two-dimensional critical fluctuation process, which consists of fields with frequencyp ₀ and has a non-Gaussian invariant distribution on these fields. Finally, we compute the critical fluctuation process in the infinite particle limit at a triple point, where a ferromagnetic and an antiferromagnetic phase transition coincide.Work supported by Deutsche Forschungsgemeinschaft     

On the detection of trends in long-term correlated records

We use the Detrended Fluctuation Analysis (DFA) to quantify underlying trends in long-term correlated records. Our approach is based on the fact that different orders of DFA are affected differently by trends. For a given instrumental record of length N, we compare the fluctuation exponent α₀ of DFA0 where trends are not being eliminated, with the fluctuation exponent α₂ of DFA2 where possible linear trends in the instrumental record are being eliminated. From this we deduce numerically the probability density p(A) that in the considered long-term correlated record, a linear trend with a slope between A and occurs. Without loss of generality we focus on Gaussian distributed data. As an example, we apply our analysis to several long temperature records (Melbourne, Oxford, Prague, Pusan, Uppsala, and Vienna), where we discuss the trends within the last 90 years, which may originate from both, urban and global warming.     

A Rigourous Demonstration of the Validity of Boltzmann’s Scenario for the Spatial Homogenization of a Freely Expanding Gas and the Equilibration of the Kac Ring

Boltzmann provided a scenario to explain why individual macroscopic systems composed of a large number N of microscopic constituents are inevitably (i.e., with overwhelming probability) observed to approach a unique macroscopic state of thermodynamic equilibrium, and why after having done so, they are then observed to remain in that state, apparently forever. We provide here rigourous new results that mathematically prove the basic features of Boltzmann’s scenario for two classical models: a simple boundary-free model for the spatial homogenization of a non-interacting gas of point particles, and the well-known Kac ring model. Our results, based on concentration inequalities that go back to Hoeffding, and which focus on the typical behavior of individual macroscopic systems, improve upon previous results by providing estimates, exponential in N, of probabilities and time scales involved.     

Stochastic theory of nonlinear rate processes with multiple stationary states

A comparison has been made between the deterministic and stochastic (master equation) formulation of nonlinear chemical rate processes with multiple stationary states. We have shown, via two specific examples of chemical reaction schemes, that the master equations have quasi-stationary state solutions which agree with the various initial condition dependent equilibrium solutions of the deterministic equations. The presence of fluctuations in the stochastic formulation leads to true equilibrium solutions, i.e. solutions which are independent of initial conditions as t → ∞. We show that the stochastic formulation leads to two distinct time scales for relaxation. The mean time for the reaction system to reach the quasi-stationary states from any initial state is of $\text{O}$(N⁰) where N is a measure of the size of the reaction system. The mean time for relaxation from a quasi-stationary state to the true equilibrium state is $\text{O}$(e^N). The results obtained from the stochastic formulation as regards the number and location of the quasi-stationary states are in complete agreement with the deterministic results.     

Study of gold nanocluster sputtering under 38-keV Au ion bombardment by a classical molecular dynamics method

The computer simulation of the interaction of 38-keV Au₁ ions with isolated spherical Au _N nanoclusters of diameters 2.6 and 18 nm is performed in the framework of the classical molecular dynamics (MD) method. The distribution of the absorbed energy ε per one atom of the irradiated cluster and the sputtering yields are analyzed for different ratios of the nanocluster diameter D to the average projective range R _p of the bombarding ion. It is established that the small values of the absorbed energy (ε ? ε_max = E/N) are most probable for D < R _p, and either small (ε ? ε_max) or the maximum possible (ε ～ ε_max) values are mainly realized for D ≤ R _p. It is shown that the total sputtering yield depends weakly on the impact parameter. It is demonstrated for the first time that the irradiated cluster, as a whole, can be ejected by direct impact with a probability of approximately 6–13%. Such events are realized in the cases where the bombarding ion causes secondary cluster-atom emission in the dominant direction to a substrate, with the result that an unsputtered cluster fraction acquires momentum in the opposite direction. This recoil effect can be one of the mechanisms for desorption of nanoclusters deposited on the surface under ion (or cluster) bombardment.     

Work distribution function for a Brownian particle driven by a nonconservative force

We derive the distribution function of work performed by a harmonic force acting on a uniformly dragged Brownian particle subjected to a rotational torque. Following the Onsager and Machlup’s functional integral approach, we obtain the transition probability of finding the Brownian particle at a particular position at time t given that it started the journey from a specific location at an earlier time. The difference between the forward and the time-reversed form of the generalized Onsager-Machlup’s Lagrangian is identified as the rate of medium entropy production which further helps us develop the stochastic thermodynamics formalism for our model. The probability distribution for the work done by the harmonic trap is evaluated for an equilibrium initial condition. Although this distribution has a Gaussian form, it is found that the distribution does not satisfy the conventional work fluctuation theorem.     

On the dynamic mean field theory of spin glasses

We derive in detail Sompolinsky's mean field theory of spin glasses using a diagram expansion of the effective local Langevin equation of Sompolinsky and Zippelius. We use a simpler generating functional than in the literature, on which the quenched average is very easily done. We pay special attention to the existence of an external field. We show that there are two different types of singularities for ω=0 in the equations. The first type, which leads to Parisi'sq(0), is connected with the local magnetisation. The second type, which leads toq′(x), is connected with the nonergodic behaviour. We show that the continuous limit of discrete Sompolinsky solutions has to be taken in order to be in accordance with the fluctuation dissipation theorem on infinite time scales. We discuss carefully the question of dynamical stability. We show that Sommers' solution is unstable only on an infinite time scale and thus remains an acceptable equilibrium theory with a broken symmetry. We argue that for ω=0 a formal violation of the fluctuation dissipation theorem is physically expected if the relaxation times are of the order of the switching time of the external field. From this point of view the spin-glass state is a steady state but not a real equilibrium state.     

Metastability and Low Lying Spectra¶in Reversible Markov Chains

We study a large class of reversible Markov chains with discrete state space and transition matrix P _N . We define the notion of a set of metastable points as a subset of the state space Γ _N such that (i) this set is reached from any point x∈Γ _N without return to x with probability at least b _N , while (ii) for any two points x, y in the metastable set, the probability T ^{− 1} _{x , y} to reach y from x without return to x is smaller than a _{N − 1}< b _N . Under some additional non-degeneracy assumption, we show that in such a situation: (i) To each metastable point corresponds a metastable state, whose mean exit time can be computed precisely. (ii) To each metastable point corresponds one simple eigenvalue of 1 −P _N which is essentially equal to the inverse mean exit time from this state. Moreover, these results imply very sharp uniform control of the deviation of the probability distribution of metastable exit times from the exponential distribution. Received: 1 August 2000 / Accepted: 19 November 2001     

Spin dynamics in crystalline and amorphous DyAg

Crystalline (cr-) Dyag [CsC] structure] orders antiferro-magnetically with T_N≅60K; amorphous (am-) DyAg ferro-magnetically with T_C≈-18K. We measured the longitudinal field (LF) μ⁺SR relaxation functions G_ZZ(t) for 5K<T<300K using surface muons. In the paramagnetic state. cr-DyAg gives an exponential G_ZZ (t) in the relaxation rate rising first slowly then more rapidlynear T_N; no decoupling is observed in LF up to 0.4T. In the ordered state we see a Lorentzian Kubo-Toyabe G_ZZ(t), becoming nearly static at the lowest temperatures. Its static width is very narrow (Δ≈-7 MHz), and full dceoupling is achieved here in 0.1 T. On approaching T_N, the fluctuation rate and the static width increase mootonically bt the field distribution remains Loratzian. A LF of 0.4T is then insufficient to quench the fast exponential relaxation. In paramagnetic am-DyAg, the μ⁺ depolarization is always much faster then in cr-DyAg. At lower temperatures it is better described by a root-exponential than an exponential G_ZZ(t). Below T_C an exponentially relaxing signal with 1/3 amplitude is seen. The decoupling effect of LF up to 0.4T was negligible at all temperatures.     

Characterization of chaotic maps using the permutation Bandt-Pompe probability distribution

By appealing to a long list of different nonlinear maps we review the characterization of time series arising from chaotic maps. The main tool for this characterization is the permutation Bandt-Pompe probability distribution function. We focus attention on both local and global characteristics of the components of this probability distribution function. We show that forbidden ordinal patterns (local quantifiers) exhibit an exponential growth for pattern-length range 3 ≤ D ≤ 8, in the case of finite time series data. Indeed, there is a minimum D _min-value such that forbidden patterns cannot appear for D < D _min. The system’s localization in an entropy-complexity plane (global quantifier) displays typical specific features associated with its dynamics’ nature. We conclude that a more “robust” distinction between deterministic and stochastic dynamics is achieved via the present time series’ treatment based on the global characteristics of the permutation Bandt-Pompe probability distribution function.     

Robust and accurate filtered spherical harmonics expansions for radiative transfer

We present a novel application of filters to the spherical harmonics (P_N) expansion for radiative transfer problems in the high-energy-density regime. The filter we use is based on non-oscillatory spherical splines and a filter strength chosen to (i) preserve the equilibrium diffusion limit and (ii) vanish as the expansion order tends to infinity. Our implementation is based on modified equations that are derived by applying the filter after every time step in a simple first-order time integration scheme. The method is readily applied to existing codes that solve the P_N equations. Numerical results demonstrate that the solution to the filtered P_N equations are (i) more robust and less oscillatory than standard P_N solutions and (ii) more accurate than discrete ordinates solutions of comparable order. In particular, the filtered P₇ solution demonstrates comparable accuracy to an implicit Monte Carlo solution for a benchmark hohlraum problem in 2D Cartesian geometry.     

Photon distribution function in blinking fluorescence of individual molecules

A photon distribution function w_N(T) for blinking fluorescence with bright on- and dark off-intervals is derived. The function w_N(T) is expressed via few Poissonian functions each of which relates to corresponding exponential process in quantum dynamics of a given individual molecule. The distribution of photons is calculated for short, middle and long time intervals as compared to off-intervals. The distributions are much broader than Poissonian distribution and have rather complicated shape. If time resolution of an experiment does not permit us to see off-interval and, therefore, fluorescence looks like CW emission, the distribution of photons gives a signal about existence of hidden off- intervals in such CW fluorescence.     

A Generalization of Wigner’s Law

We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions , with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p _N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p _N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k ^th moment of p _N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p _N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.