On the probability distribution of stock returns in the Mike-Farmer model

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index H_s, the distribution of relative order prices x in reference to the same best price described by a Student distribution (or Tsallis’ q-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index H_s and the freedom degree α_x of the Student distribution can always produce power-law tails in the return distribution f_r(r) with different tail exponent α_r. In this paper, we study the origin of the power-law tails of the return distribution f_r(r) in the MF model, based on extensive simulations with different combinations of the left part L(x) for x < 0 and the right part R(x) for x > 0 of f_x(x). We find that power-law tails appear only when L(x) has a power-law tail, no matter R(x) has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.     

Distribution of return intervals of extreme events

The distribution of return intervals of extreme events is studied in time series characterized by finite-term correlations with non-exponential decay. Precisely, it has been analyzed the statistics of the return intervals of extreme values of the resistance fluctuations displayed by resistors with granular structure in nonequilibrium stationary states. The resistance fluctuations are calculated by Monte Carlo simulations using a resistor network approach. It has been found that for highly disordered networks, when the auto-correlation function displays a non-exponential and non-power-law decay, the distribution of return intervals of the extreme values is a stretched exponential, with exponent independent of the threshold.     

Sensitivity to initial conditions in the Bak-Sneppen model of biological evolution

We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds of that of the tuned onset of chaos in say logistic-like one-dimensional maps. We also calculate the dynamical exponent z. Received: 5 November 1997 / Received in final form: 11 November 1997 / Accepted: 19 November 1997     

Persistence in systems with algebraic interaction

Persistence in coarsening one-dimensional spin systems with a power-law interaction r(-1-sigma) is considered. Numerical studies indicate that for sufficiently large values of the interaction exponent sigma (sigma > or =1/2 in our simulations), persistence decays as an algebraic function of the length scale L, P(L) approximately L(-theta). The persistence exponent theta is found to be independent on the force exponent sigma and close to its value for the extremal (sigma-->infinity) model, theta =0.175 075 88. For smaller values of the force exponent (sigma < 1/2), finite size effects prevent the system from reaching the asymptotic regime. Scaling arguments suggest that in order to avoid significant boundary effects for small sigma, the system size should grow as [O(1/sigma)](1/sigma).     

Critical comparison of several order-book models for stock-market fluctuations

Far-from-equilibrium models of interacting particles in one dimension are used as a basis for modelling the stock-market fluctuations. Particle types and their positions are interpreted as buy and sel orders placed on a price axis in the order book. We revisit some modifications of well-known models, starting with the Bak-Paczuski-Shubik model. We look at the four decades old Stigler model and investigate its variants. One of them is the simplified version of the Genoa artificial market. The list of studied models is completed by the models of Maslov and Daniels et al. Generically, in all cases we compare the return distribution, absolute return autocorrelation and the value of the Hurst exponent. It turns out that none of the models reproduces satisfactorily all the empirical data, but the most promising candidates for further development are the Genoa artificial market and the Maslov model with moderate order evaporation.     

Island distance in one-dimensional epitaxial growth

The typical island distance in submonolayer epitaxial growth depends on the growth conditions via an exponent . This exponent is known to depend on the substrate dimensionality, the dimension of the islands, and the size i^* of the critical nucleus for island formation. In this paper we study the dependence of on i^* in one-dimensional epitaxial growth. We derive that for and confirm this result by computer simulations. Received: 26 May 1998 / Accepted: 23 June 1998     

Localization-delocalization transition in one-dimensional system with long-range correlated diagonal disorder

We show that the electronic states in a one-dimensional (1D) Anderson model of diagonal disorder with long-range correlation proposed by de Moura and Lyra exhibit localization-delocalization phase transition in varying the energy of electrons. Using transfer matrix method, we calculate the average resistivity and investigate how it changes with the size of the system N. For given value of α (> 2) we find critical energies E_c1 and E_c2 such that the resistivity decreases with N as a power law ∝ N ^{- γ} for electron energies within the range of [E _c1, E _c2], and exponentially grows with N outside this range. Such behaviors persist in approaching the transition points and the exponent γ is in the range from 0.92 to 0.96. The origin of the delocalization in this 1D model is discussed. Received 18 December 2001 / Received in final form 2 May 2002 Published online 14 October 2002 RID="a" ID="a"e-mail: sjxiong@nju.edu.cn     

Ground state and glass transition of the RNA secondary structure

RNA molecules form a sequence-specific self-pairing pattern at low temperatures. We analyze this problem using a random pairing energy model as well as a random sequence model that includes a base stacking energy in favor of helix propagation. The free energy cost for separating a chain into two equal halves offers a quantitative measure of sequence specific pairing. In the low temperature glass phase, this quantity grows quadratically with the logarithm of the chain length, but it switches to a linear behavior of entropic origin in the high temperature molten phase. Transition between the two phases is continuous, with characteristics that resemble those of a disordered elastic manifold in two dimensions. For designed sequences, however, a power-law distribution of pairing energies on a coarse-grained level may be more appropriate. Extreme value statistics arguments then predict a power-law growth of the free energy cost to break a chain, in agreement with numerical simulations. Interestingly, the distribution of pairing distances in the ground state secondary structure follows a remarkable power-law with an exponent -4/3, independent of the specific assumptions for the base pairing energies.     

Multiscaling and localized instabilities in fracture, fragmentation, and growth processes

It is shown that localized instabilities can be an origin of log-normal and power-law statistical distributions in fracture, fragmentation and island growth processes. Results of laboratory experiments and numerical simulations performed by different authors are used to demonstrate the applicability of this approach. Received 2 September 1999     

Inversion symmetry and exact critical exponents of dissipating waves in the sandpile model

By an inversion symmetry, we show that in the Abelian sandpile model the probability distribution of dissipating waves of topplings that touch the boundary of the system shows a power-law relationship with critical exponent 5/8 and the probability distribution of those dissipating waves that are also last in an avalanche has an exponent of 1. Our extensive numerical simulations not only support these predictions, but also show that inversion symmetry is useful for the analysis of the two-wave probability distributions.     

Accurate Treatment of Tunnel Ionization,Induced by a Constant Electric Field,From a Power-Law Singular Potential

The tunnel ionization, induced by a weak electrostatic field, of a particle bound in an attractive one-dimensional power-law singular potential is analysed. An accurate approximation applicable for the general case of non-fractional power exponent is suggested.     

Power, Lévy, exponential and Gaussian-like regimes in autocatalytic financial systems

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 α of these power laws are time independent and depend only on the way the elements with very small values are treated. These truncated power laws determine the collective time evolution of the system. In particular the global stochastic fluctuations of the system differ from the normal Gaussian noise according to the time and size scales at which these fluctuations are considered. We describe the ranges in which these fluctuations are parameterized respectively by: the Lévy regime α < 2, the power law decay with large exponent ( α > 2), and the exponential decay. Finally we relate these results to the large exponent power laws found in the actual behavior of the stock markets and to the exponential cut-off detected in certain recent measurement. Received 29 July 2000 and Received in final form 25 September 2000     

Numerical study of local and global persistence in directed percolation

The local persistence probability P _l (t) that a site never becomes active up to time t, and the global persistence probability P _g (t) that the deviation of the global density from its mean value does not change its sign up to time t are studied in a (1+1)-dimensional directed percolation process by Monte-Carlo simulations. At criticality, starting from random initial conditions, P _l (t) decays algebraically with the exponent . The value is found to be independent of the initial density and the microscopic details of the dynamics, suggesting is an universal exponent. The global persistence exponent is found to be equal or larger than . This contrasts with previously known cases where . It is shown that in the special case of directed-bond percolation, P _l (t) can be related to a certain return probability of a directed percolation process with an active source (wet wall). Received: 15 December 1997 / Revised: 6 April 1998 / Accepted: 29 May 1998     

Fluctuations, response and aging dynamics in a simple glass-forming liquid out of equilibrium

By means of molecular dynamics computer simulations we investigate the out of equilibrium relaxation dynamics of a simple glass former, a binary Lennard-Jones system, after a quench to low temperatures. We find that one-time quantities, such as the energy or the structure factor, show only a weak time dependence. By comparing the out of equilibrium structure factor with equilibrium data we find evidence that during the aging process the system remains in that part of phase space that mode-coupling theory classifies as liquid like. Two-times correlation functions show a strong time and waiting time dependence. For large and times corresponding to the early -relaxation regime the correlators approach the Edwards-Anderson value by means of a power-law in time. For large but fixed values of the relaxation dynamics in the -relaxation regime seems to be independent of the observable and temperature. The -relaxation shows a power-law dependence on time with an exponent which is independent of but depends on the observable. We find that at long times the correlation functions can be expressed as and compute the function h(t). This function is found to show a t-dependence which is a bit stronger than a logarithm and to depend on the observable considered. If the system is quenched to very low temperatures the relaxation dynamics at long times shows fast drops as a function of time. We relate these drops to relatively local rearrangements in which part of the sample relaxes its stress by a collective motion of 50-100 particles. Finally we discuss our measurements of the time dependent response function. We find that at long times the correlation functions and the response are not related by the usual fluctuation dissipation theorem but that this relation is similar to the one found for spin glasses with one step replica symmetry breaking. Received 17 May 1999     

Self-affine random surfaces

We consider general d-dimensional random surfaces that are characterized by power-law power spectra defined in both infinite and finite spectral regions. The first type of surfaces belongs to the class of ideal fractals, whereas the second possess both the smallest and the largest scales and physically is more realistic. For both types we calculate the structure functions (SF) exactly; in addition for the second type we obtain the SF's asymptotic expansions. On this basis we show that the surfaces are (in statistical sense) self-affine and approximately self-affine, respectively. Depending on the value of the spectral exponent, we find imbalance between the finite size effects which results in systematic discrepancy in the scaling properties between the two types of surfaces. Explicit expressions for the topothesy, and in the case of second type of surfaces for the large correlation length and cross-over distances are also derived. Received 3 October 2001 / Received in final form 5 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: oyordanov@aubg.bg     

Phase transition in annihilation-limited processes

A system of particles is studied in which the stochastic processes are one-particle type-change (or one-particle diffusion) and multi-particle annihilation. It is shown that, if the annihilation rate tends to zero but the initial values of the average number of the particles tend to infinity, so that the annihilation rate times a certain power of the initial values of the average number of the particles remain constant (the double scaling) then if the initial state of the system is a multi-Poisson distribution, the system always remains in a state of multi-Poisson distribution, but with evolving parameters. The large time behavior of the system is also investigated. The system exhibits a dynamical phase transition. It is seen that for a k-particle annihilation, if k is larger than a critical value k_c, which is determined by the type-change rates, then annihilation does not enter the relaxation exponent of the system; while for k < k_c, it is the annihilation (in fact k itself) which determines the relaxation exponent.     

Damage spreading in 2-dimensional isotropic and anisotropic Bak-Sneppen models

We implement the damage spreading technique on 2-dimensional isotropic and anisotropic Bak-Sneppen models. Our extensive numerical simulations show that there exists a power-law sensitivity to the initial conditions at the statistically stationary state (self-organized critical state). Corresponding growth exponent α for the Hamming distance and the dynamical exponent z are calculated. These values allow us to observe a clear data collapse of the finite size scaling for both versions of the Bak-Sneppen model. Moreover, it is shown that the growth exponent of the distance in the isotropic and anisotropic Bak-Sneppen models is strongly affected by the choice of the transient time.     

Pinning of cracks in two-dimensional disordered media

We study the statistics of crack pinning in two dimensions by experiments and simulations of directed polymers in random media. Mode I tensile tests on paper samples show a delocalization phenomenon as the notch length is varied if the fraction of cracks pinned to the notch is monitored. This is compared with the behavior of directed polymers in the presence of both an energetically favorable localized pinning center and bulk disorder. An analysis of the crack interface roughness indicates self-affine behavior with a roughness exponent ζ in the proximity of the minimum energy surface value 2/3. Received 4 April 2000 and Received in final form 10 October 2000     

Cooperative dynamics of snowdrift game on spatial distance-dependent small-world networks

We investigate the evolution of cooperative behaviors of small-world networking agents in a snowdrift game mode, where two agents (nodes) are connected with probability depending on their spatial Euclidean lattice distance in the power-law form controlled by an exponent α. Extensive numerical simulations indicate that the game dynamics crucially depends on the spatial topological structure of underlying networks with different values of the exponent α. Especially, in the distance-independent case of α=0, the small-world connectivity pattern contributes to an enhancement of cooperation compared with that in regular lattices, even for the case of having a high cost-to-benefit ratio r. However, with the increment of α>0, when r≥0.4, the spatial distance-dependent small-world (SDSW) structure tends to inhibit the evolution of cooperation in the snowdrift game.