Narrow Results

Recent progress towards an understanding of fluctuational escape from chaotic attractors (CAs) is reviewed and discussed in the contexts of both continuous systems and maps. It is shown that, like the simpler case of escape from a regular attractor, a unique most probable escape path (MPEP) is followed from a CA to the boundary of its basin of attraction. This remains true even where the boundary structure is fractal. The importance of the boundary conditions on the attractor is emphasized. It seems that a generic feature of the escape path is that it passes via certain unstable periodic orbits. The problems still remaining to be solved are identified and considered.

The statistical properties of sequences of Waiting Times generated by Non-Homogeneous Poisson Processes are investigated by means of Renewal Aging, i.e. a statistical analysis being able to detect the presence of genuine renewal events with non-Poisson statistics. The Renewal Aging features of two non-homogeneous Poisson models with different prescriptions of the rate modulation are compared. Both prescriptions are essentially periodic. In the first model, the rate is a totally deterministic and smooth harmonic function. In the second model, the rate modulation is deterministic and smooth almost everywhere, except in some singular points occurring with a periodicity affected by weak random fluctuations.

The main parameter of both models is the modulation speed, defined as the ratio between the modulation period and the average Waiting Time. This allows to distinguish between slow, intermediate and fast modulation of the rate. As expected, in the very slow modulation case both models have a strongly reduced Renewal Aging, practically zero, because the analysis is affected by local clusters of Waiting Times with the same Poisson rate. In the intermediate range of modulation speeds, both models show non-zero Renewal Aging and the behavior of the two models tends to be more different as the modulation speed decreases.

Going towards the fast modulation regime, the behavior of the two models becomes completely different from one another. Model I does not show a definite global Renewal Aging. On the time scale of the modulation period, local oscillations arise in the Probability Density Function and an apparent oscillating aging is also observed. However, this form of aging is not the manifestation of genuine renewal non-Poisson events, but it is related to the local oscillations. On the contrary, Model II displays Renewal Aging features in agreement with a homogeneous Renewal non-Poisson Process. It is argued that this is related to the singular points in the rate modulation of Model II, which, due to their random occurrence, could be identified with genuine critical non-Poisson events. The approach presented here could have interesting applications in problems involving dichotomous noise and, in particular, in the statistical characterization of ON–OFF fluorescence intermittency experimentally observed in complex systems, such as nano-crystals and single bio-molecules stimulated by laser fields.

We study the effect of external noise on the catalytic oxidation of CO on an Iridium(111) single crystal under ultrahigh vacuum conditions. This reaction can be considered as a model of catalysis used in the industry. In the absence of noise, the reaction exhibits one or two stable stationary states, depending on the control parameters such as temperature and partial pressures. When noise is added, for instance, by randomly varying the quality of the influx mixture, the system exhibits stochastic reaction rate and switching. In this work, we present the theoretical predictions for the bistable situation using a number of colored-noise approximations that consider the intensity and correlation of the external fluctuations. As we will show, only some of these approximations are suitable for bistable systems.

The recent turbulence on the world's stock markets has reinvigorated the attack on classical economic models of stock market fluctuations. The key problem is determining a dynamic model, which is consistent with observed fluctuations and which reflects investor behavior. Here, we use a novel equation-free approach developed in nonlinear dynamics literature to identify the salient statistical features of fluctuations of the Dow Jones Industrial Average over the past 80 years. We then develop a minimal dynamical model in the form of a stochastic differential equation involving both additive and multiplicative system-noise couplings, which captures these features and whose parameterization on a time scale of days can be used to capture market distributions up to a time scale of months. The terms in the model can be directly linked to "herding" behavior on the part of traders. However, we show that parameters in this model have changed over a number of decades producing different market regimes. This result partially explains how, during some periods of history, "classic" economic models may work well and at other periods "econo-physics" models prove better.

Global coordination for the preservation of a common good, such as climate, is one of the most prominent challenges of modern societies. In this manuscript, we use the framework of evolutionary game theory to investigate whether a polycentric structure of multiple small-scale agreements provides a viable solution to solve global dilemmas as climate change governance. We review a stochastic model which incorporates a threshold game of collective action and the idea of risky goods, capturing essential features unveiled in recent experiments. We show how reducing uncertainty both in terms of the perception of disaster and in terms of goals induce a transition to cooperation. Taking into account wealth inequality, we explore the impact of the homophily, potentially present in the network of influence of the rich and the poor, in the different contributions of the players. Finally, we discuss the impact of polycentric sanctioning institutions, showing how such a scenario also proves to be more efficient than a single global institution.

The aim of this work is to study a stochastic individual-based model, structured with respect to age (progression within the cell cycle) and space (radial distance from the oocyte). We prove the existence of solutions and the convergence in large population and size scale limit to a solution of a partial differential equation.

We study the dynamics of a particle in a space that is non-differentiable. Non-smooth geometrical objects have an inherently probabilistic nature and, consequently, introduce stochasticity in the motion of a body that lives in their realm. We use the mathematical concept of fiber bundle to characterize the multivalued nature of geodesic trajectories going through a point that is non-differentiable. Then, we generalize our concepts to everywhere non-smooth structures. The resulting theoretical framework can be considered a hybridization of the theory of surfaces and the theory of stochastic processes. We keep the concepts as general as possible, in order to allow for the introduction of other fundamental processes capable of modeling the fractality or the fluctuations of any conceivable continuous, but non-differentiable space.

In this work, we study the multifractal properties of the return intervals between fading events defined as laser intensity drops induced by optical turbulence. A laboratory-controlled experiment was conducted in which the irradiance fluctuations were recorded at a high sampling rate. Return intervals were obtained by identifying fading events under a threshold defined in times of the standard deviation of the measured irradiance. Using multifractal detrended fluctuation analysis, we found a crossover between two different scaling regimes: a spurious multifractal slightly correlated at small scales and a monofractal correlated for large scales. Based on a surrogate analysis and numerical simulations of monofractal processes, we conclude the heavy tail of the probability distribution of the return interval is the origin of the multifractality observed at small scales.

Non-homogeneous Poisson process (NHPP) software reliability growth models (SRGM^a) enable quantitative metrics to guide decisions during the software engineering life cycle, including test resource allocation and release planning. However, many SRGM possess complex mathematical forms that make them difficult to apply. Specifically, traditional procedures solve a system of nonlinear equations to identify the numerical parameters that best characterize failure data. Recently, researchers have developed expectation-maximization (EM) algorithms for NHPP SRGM that exhibit better convergence properties and can therefore find maximum likelihood estimates with greater ease.

This paper presents an adaptive EM (AEM) algorithm, which combines an earlier EM algorithm for NHPP SRGM with unconstrained search of the model parameter space. Our performance analysis shows that the AEM outperforms state-of-the-art EM algorithms for NHPP SRGM with very strong statistical significance, which is as much as hundreds of times faster on some data sets. Thus, the approach can fit SRGM very quickly. We also incorporate this high performance adaptive EM algorithm into a heuristic nested model selection procedure to objectively select a model of least complexity that best characterizes the failure data. Results indicate this heuristic approach often identifies the model possessing the best model selection criteria.

Acronyms are not pluralized.

We investigate the effect of stochastic fluctuations of an interest rate on the value of a derivative. We derive the modified Black-Scholes equation that describes evolution of the value of a derivative averaged over an ensemble of stochastic fluctuations of the rate of interest and depends on the "renormalized" values of volatility and rate of interest. We present the explicit expressions for the renormalized volatility and interest rate that incorporate the corrections owing to the short-term stochastic variations of the interest rate. The stochastic component of the interest rate tends to enhance the effective volatility and reduce the effective interest rate that determine an evolution of the option pricing "smoothed out" over the stochastic variations. The results of numerical solution of the modified Black-Scholes equation with the renormalized coefficients are illustrated for an American put option on non-dividend-paying stock.

A computational technique borrowed from the physical sciences is introduced to obtain accurate closed-form approximations for the transition probability of arbitrary diffusion processes. Within the path integral framework the same technique allows one to obtain remarkably good approximations of the pricing kernels of financial derivatives. Several examples are presented, and the application of these results to increase the efficiency of numerical approaches to derivative pricing is discussed.

Bidirectional valuation models are based on numerical methods to obtain kernels of parabolic equations. Here we address the problem of robustness of kernel calculations vis a vis floating point errors from a theoretical standpoint. We are interested in kernels of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step h > 0 in the limit as h → 0. We consider both semidiscrete triangulations with continuous time and explicit Euler schemes with time step so small that the Courant condition is satisfied. We find uniform bounds for the convergence rate as a function of the degree of smoothness. We conjecture these bounds are indeed sharp. The bounds also apply to the time derivatives of the kernel and its first two space derivatives. The proof is constructive and is based on a new technique of path conditioning for Markov chains and a renormalization group argument. We make the simplifying assumption of time-independence and use longitudinal Fourier transforms in the time direction. Convergence rates depend on the degree of smoothness and Hölder differentiability of the coefficients. We find that the fastest convergence rate is of order O(h²) and is achieved if the coefficients have a bounded second derivative. Otherwise, explicit schemes still converge for any degree of Hölder differentiability except that the convergence rate is slower. Hölder continuity itself is not strictly necessary and can be relaxed by an hypothesis of uniform continuity.

We present an accurate and easy-to-compute approximation of zero-coupon bonds and Arrow–Debreu (AD) prices for the Black–Karasinski (BK) model of interest rates or default intensities. Through this procedure, dubbed exponent expansion, AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results — for time horizons up to several years — even when truncated to the first few terms. For larger time horizons the exponent expansion can be combined with a fast numerical convolution to obtain extremely accurate results.

We derive semi-analytic approximation formulae for bond and swaption prices in a Black–Karasiński (BK) interest rate model. Approximations are obtained using a novel technique based on the Karhunen–Loève expansion. Formulas are easily computable and prove to be very accurate in numerical tests. This makes them useful for numerically efficient calibration of the model.

In this work, we present an analytical model, based on the path-integral formalism of statistical mechanics, for pricing options using first-passage time problems involving both fixed and deterministically moving absorbing barriers under possibly non-Gaussian distributions of the underlying object. We adapt to our problem a model originally proposed by De Simone et al. (2011) to describe the formation of galaxies in the universe, which uses cumulant expansions in terms of the Gaussian distribution, and we generalize it to take into account drift and cumulants of orders higher than three. From the probability density function, we obtain an analytical pricing model, not only for vanilla options (thus removing the need of volatility smile inherent to the Black & Scholes (1973) model), but also for fixed or deterministically moving barrier options. Market prices of vanilla options are used to calibrate the model, and barrier option pricing arising from the model is compared to the price resulted from the relative entropy model.

For stochastic processes of non-commuting random variables, we formulate a Cox–Ingersoll–Ross (CIR) stochastic differential equation in the context of free probability theory which was introduced by D. Voiculescu. By transforming the classical CIR equation and the Feller condition, which ensures the existence of a positive solution, into the free setting (in the sense of having a strictly positive spectrum), we show the global existence for a free CIR equation. The main challenge lies in the transition from a stochastic differential equation driven by a classical Brownian motion to a stochastic differential equation driven by the free analogue to the classical Brownian motion, the so-called free Brownian motion.

The features of 1/f-noise processes offer important new insights into the field of population biology, greatly helping our quest for understanding and for prediction of ecological processes. 1/f-noises account quite satisfactorily for the observed nature of ecological fluctuations. This article reviews the application of 1/f-noise processes to ecology. After a discussion of the basic problems of population ecology that makes such an innovation necessary, we review the features of 1/f-noises concentrating especially on those aspects that make these processes attractive as a solution. We also present a discussion of the analysis of real ecological data, which confirms that there are good empirical as well as good theoretical reasons to establish a leading role for pink 1/f noise. We then discuss the consequences of such a model for our understanding of ecology. The article finishes with a number of observations about some aspects of ecological data and applications that are likely to drive research in a different direction from that associated with engineering and the physical sciences.

We focus on the stochastic description of the stock price dynamics. Thereby we concentrate on the Heston model and the Hull–White model. We derive the stationary probability density distribution of the variance of both models in the case of zero correlation coefficient. These distributions are used to calculate solutions for the logarithmic returns of the stock price for short time lags. Furthermore we apply the solutions of both models to the German tick-by-tick Dax data [1]. The data are from May 1996 to December 2001. We use the probability density distributions of the logarithmic returns, calculated out of the data, and fit these distributions to the theoretical distributions.

We investigate 738 complete genomes of viruses to detect the presence of short inverted repeats. The number of inverted repeats found is compared with the prediction obtained for a Bernoullian and for a Markovian control model. We find as a statistical regularity that the number of observed inverted repeats is often greater than the one expected in terms of a Bernoullian or Markovian model in several of the viruses and in almost all those with a genome longer than 30,000 bp.

The effect caused by the presence of a number of distinct time scales in a simple stochastic model for the Earth's atmosphere temperature fluctuations is studied. The model is described by a dissipative dynamics consisting of a set of coupled stochastic evolution equations. The system shows features that resemble recent observations. In contrast to other approaches, like autoregressive models, the fluctuations of the atmosphere's temperature depend on parameters with clear physical meaning. A reduced version of the model is constructed and its temporal autocovariance function is explicitly written.