Volatility clustering and scaling for financial time series due to attractor bubbling

A microscopic model of financial markets is considered, consisting of many interacting agents (spins) with global coupling and discrete-time thermal bath dynamics, similar to random Ising systems. The interactions between agents change randomly in time. In the thermodynamic limit the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on-off intermittency, resembling the empirical financial time series with volatility clustering. For a proper choice of the model parameters the probability distributions of returns exhibit power-law tails with scaling exponents close to the empirical ones.Comment: For related publications see http://www.helbing.or

Numerical integration of the stochastic Landau-Lifshitz-Gilbert equation in generic time-discretisation schemes

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.Comment: 9 pages and 7 figure

Phase transitions induced by microscopic disorder: a study based on the order parameter expansion

Based on the order parameter expansion, we present an approximate method which allows us to reduce large systems of coupled differential equations with diverse parameters to three equations: one for the global, mean field, variable and two which describe the fluctuations around this mean value. With this tool we analyze phase-transitions induced by microscopic disorder in three prototypical models of phase-transitions which have been studied previously in the presence of thermal noise. We study how macroscopic order is induced or destroyed by time independent local disorder and analyze the limits of the approximation by comparing the results with the numerical solutions of the self-consistency equation which arises from the property of self-averaging. Finally, we carry on a finite-size analysis of the numerical results and calculate the corresponding critical exponents

Prime numbers and spontaneous neuron activity

Logarithmic gaps have been used in order to find a periodic component of the sequence of prime numbers, hidden by a random noise (stochastic or chaotic). It is shown that multiplicative nature of the noise is the main reason for the successful application of the logarithmic gaps transforming the multiplicative noise into an additive one. A relation of this phenomenon to spontaneous neuron activity and to chaotic brain computations has been discussed.Comment: extende

Robust filtering for bilinear uncertain stochastic discrete-time systems

Copyright [2002] IEEE. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of Brunel University's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to [email protected]. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This paper deals with the robust filtering problem for uncertain bilinear stochastic discrete-time systems with estimation error variance constraints. The uncertainties are allowed to be norm-bounded and enter into both the state and measurement matrices. We focus on the design of linear filters, such that for all admissible parameter uncertainties, the error state of the bilinear stochastic system is mean square bounded, and the steady-state variance of the estimation error of each state is not more than the individual prespecified value. It is shown that the design of the robust filters can be carried out by solving some algebraic quadratic matrix inequalities. In particular, we establish both the existence conditions and the explicit expression of desired robust filters. A numerical example is included to show the applicability of the present method

Algebraic coarsening in voter models with intermediate states

The introduction of intermediate states in the dynamics of the voter model modifies the ordering process and restores an effective surface tension. The logarithmic coarsening of the conventional voter model in two dimensions is eliminated in favour of an algebraic decay of the density of interfaces with time, compatible with Model A dynamics at low temperatures. This phenomenon is addressed by deriving Langevin equations for the dynamics of appropriately defined continuous fields. These equations are analyzed using field theoretical arguments and by means of a recently proposed numerical technique for the integration of stochastic equations with multiplicative noise. We find good agreement with lattice simulations of the microscopic model.Comment: 11 pages, 5 figures; minor typos correcte

System Size Stochastic Resonance: General Nonequilibrium Potential Framework

We study the phenomenon of system size stochastic resonance within the nonequilibrium potential's framework. We analyze three different cases of spatially extended systems, exploiting the knowledge of their nonequilibrium potential, showing that through the analysis of that potential we can obtain a clear physical interpretation of this phenomenon in wide classes of extended systems. Depending on the characteristics of the system, the phenomenon results to be associated to a breaking of the symmetry of the nonequilibrium potential or to a deepening of the potential minima yielding an effective scaling of the noise intensity with the system size.Comment: LaTex, 24 pages and 9 figures, submitted to Phys. Rev.