Weiner Process Research Articles

Finite-time stochastic combination synchronization of three different chaotic systems and its application in secure communication

In this paper, the finite-time stochastic combination synchronization of three different chaotic systems is investigated. Based on the adaptive technique and the properties of Weiner process, a novel sufficient condition is obtained to ensure combination synchronization under stochastic perturbations. Moreover, a secure communication scheme based on the adaptive combination synchronization of three different systems, i.e., the Lorenz system, Chen system, and Lü system, with uncertainties, unknown parameters, and stochastic perturbation is presented. The simulation results show the feasibility of the proposed method.

Stochastic synchronization for time-varying complex dynamical networks

This paper studies the stochastic synchronization problem for time-varying complex dynamical networks. This model is totally different from some existing network models. Based on the Lyapunov stability theory, inequality techniques, and the properties of the Weiner process, some controllers and adaptive laws are designed to ensure achieving stochastic synchronization of a complex dynamical network model. A sufficient synchronization condition is given to ensure that the proposed network model is mean-square stable. Theoretical analysis and numerical simulation fully verify the main results.

Estimation of Equicorrelated Diffusions from Incomplete Data

The paper derives maximum likelihood parameter estimators for symmetrically correlated Weiner processes observed at discrete intervals. Such processes arise when pricing and determining Value-at-Risk for portfolio derivatives. Cases of driftless and mean-reverting state variables are considered. The procedure is applicable to samples with missing data of any pattern and to high dimensional systems. The estimation procedure is illustrated using a sample of stock prices.

On Durbin's Series for the Density of First Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

On Durbin's Series for the Density of First Passage Times

Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

Stochastic quasi-synchronization for delayed dynamical networks via intermittent control

In this paper, we consider the stochastic quasi-synchronization for the delayed networks with parameter mismatches and stochastic perturbation mismatch by using intermittent control. Based on Lyapunov stability theory, inequality techniques and the properties of Weiner process, several sufficient conditions are obtained to ensure stochastic quasi-synchronization for delayed networks. Meanwhile, numerical simulations are offered to show the effectiveness of our new results.

Finite-time stochastic synchronization of complex networks

In this paper, we study the finite-time stochastic synchronization problem for complex networks with stochastic noise perturbations. By using finite-time stability theorem, inequality techniques, the properties of Weiner process and adding suitable controllers, sufficient conditions are obtained to ensure finite-time stochastic synchronization for the complex networks. The effects of control parameters on synchronization speed and time are also analyzed. The results of this paper are applicable to both directed and undirected weighted networks while do not need to know any information about eigenvalues of coupling matrix. Since finite-time synchronization means the optimality in convergence time and has better robustness and disturbance rejection properties, the results of this paper are important. A numerical example shows the effectiveness of our new results.

Futures price modeling under exchange rate volatility and its multi-period semi-variance portfolio selection

Considering the stochastic exchange rate, a four-factor futures model with the underling asset, convenience yield, instantaneous risk-free interest rate and exchange rate, is established. These processes follow jump-diffusion processes (Weiner process and Poisson process). The corresponding partial differential equation (PDE) with terminal boundary condition of the model is drawn. The general solution with parameters of the above PDE is derived. The parameters are estimated by using the weight least squares approach with historical data for special cases. For the objective of risk assessment, downside risk has impacted on the practitioner's view of risk apparently. Variance is substituted by semi-variance. Moreover, one period portfolio selection is extended to multi-period. A class of multi-period semi-variance model is formulated. A hybrid genetic algorithm, which makes use of the position displacement strategy of the particle swarm optimiser as a mutation operation, is applied to solve the multi-period semi-variance model. Finally, in order to demonstrate the effectiveness of the theoretical models and numerical methods, fuel futures in the Shanghai exchange market is selected to be an example.

Stochastic synchronization of coupled neural networks with intermittent control

In this Letter, we study the exponential stochastic synchronization problem for coupled neural networks with stochastic noise perturbations. Based on Lyapunov stability theory, inequality techniques, the properties of Weiner process, and adding different intermittent controllers, several sufficient conditions are obtained to ensure exponential stochastic synchronization of coupled neural networks with or without coupling delays under stochastic perturbations. These stochastic synchronization criteria are expressed in terms of several lower-dimensional linear matrix inequalities (LMIs) and can be easily verified. Moreover, the results of this Letter are applicable to both directed and undirected weighted networks. A numerical example and its simulations are offered to show the effectiveness of our new results.

A class of portfolio selection with a four-factor futures price model

Considering the stochastic exchange rate, a four-factor futures model with the underling asset, convenience yield, instantaneous risk free interest rate and exchange rate, is established. These processes follow jump-diffusion processes (Weiner process and Poisson process). The corresponding partial differential equation (PDE) of the futures price is derived. The general solution of the PDE with parameters is drawn. The weight least squares approach is applied to obtain the parameters of above PDE. Variance is substituted by semi-variance in Markowitzs portfolio selection model. Therefore, a class of multi-period semi-variance model is formulated originally. Then, a continuous-time mean-variance portfolio model is also considered. The corresponding stochastic Hamilton-Jacobi-Bellman (HJB) equation of the problem with nonlinear constraints is derived. A numerical algorithm is proposed for finding the optimal solution in this paper. Finally, in order to demonstrate the effectiveness of the theoretical models and numerical methods, the fuel futures in Shanghai exchange market and the Brent crude oil futures in London exchange market are selected to be examples.

Adaptive synchronization of two chaotic systems with stochastic unknown parameters

Using the Lyapunov stability theory an adaptive control is proposed for chaos synchronization between two different systems which have stochastically time varying unknown coefficients. The stochastic variations of the coefficients about their unknown mean values are modeled through white Gaussian noise produced by the Weiner process. It is shown that using the proposed adaptive control the mean square of synchronization error converges to an arbitrarily small bound around zero. To demonstrate the effectiveness of the proposed technique, it is applied to the Lorenz–Chen and the Chen–Rossler dynamical systems, as some case studies. Simulation results indicate that the proposed adaptive controller has a high performance in synchronization of chaotic systems in noisy environment.

Symmetries of first‐order stochastic ordinary differential equations revisited

AbstractSymmetries of stochastic ordinary differential equations (SODEs) are analysed. This work focuses on maintaining the properties of the Weiner processes after the application of infinitesimal transformations. The determining equations (DEs) for first‐order SODEs are derived in an Itô calculus context. These DEs are non‐stochastic. This article reconciles earlier works in this area. Copyright © 2007 John Wiley & Sons, Ltd.

Modeling and inversion of net ecological exchange data using an Ito stochastic differential equation approach

A system of stochastic differential equations is studied describing a compartmental carbon transfer model that includes uncertainties arising in the model from environmental and photosynthetic effects as well as initial conditions. Justification is given for the modeling of observed times series as Weiner processes. The solution of the resulting system is obtained as a stochastic process, a formulation is given appropriate for obtaining continuity and differentiability with respect to model transfer coefficients, and numerical approximation results are given. Estimation results of coefficients from NEE data are obtained using quasi-Monte Carlo techniques. The error resulting in NEE stochastic models is observed to be approximately Gaussian. This result is used to construct a joint probability density defined on a sample space of transfer coefficients. Finally, the joint probability density function is used with quasi-Monte Carlo to obtain information on transfer coefficients and predicted carbon pools. This information is compared with a priori results obtained without the benefit of NEE data.

Adaptive control of chaotic systems with stochastic time varying unknown parameters

In this paper based on the Lyapunov stability theorem, an adaptive control scheme is proposed for stabilizing the unstable periodic orbits (UPO) of chaotic systems. It is assumed that the chaotic system has some linearly dependent unknown parameters which are stochastically time varying. The stochastic parameters are modeled through the Weiner process derivative. To demonstrate the effectiveness of the proposed technique it has been applied to the Lorenz, Chen and Rossler dynamical systems, as some case studies. Simulation results indicate that the proposed adaptive controller has a high performance in stabilizing the UPO of chaotic systems in noisy environment.

On some stochastic parabolic differential equations in a Hilbert space

We consider some stochastic difference partial differential equations of the form du(x, t, c) = L(x, t, D)u(x, t, c)dt + M(x, t, D)u(x, t − a, c)dw(t), where L(x, t, D) is a linear uniformly elliptic partial differential operator of the second order, M(x, t, D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.

A Vitality-Based Model Relating Stressors and Environmental Properties to Organism Survival

A survivorship curve is shaped by the differential survivability of the organisms within the population, and a change in a survivorship curve with a stressor reflects the differential response of the organisms to the stressor. Quantifying this linkage in a simple, rigorous way is valuable for characterizing the response of populations to stressors and ultimately for understanding the evolutionary selection of individuals exposed to stressors. To quantify this stressor–individual–population linkage with as few parameters as possible, I present a simple mechanistic model describing organism survival in terms of age-dependent and age-independent mortality rates. The age-independent rate is represented by a Poisson process. For the age-dependent rate, a concept of vitality is defined, and mortality occurs when an organism's vitality is exhausted. The loss of vitality over age is represented by a continuous Brownian-motion process, the Weiner process; vitality-related mortality occurs when the random process reaches the boundary of zero vitality. The age at which vitality-related mortality occurs is represented by the Weiner-process probability distribution for first-arrival time. The basic model has three rate parameters: the rate of accidental mortality, the mean rate of vitality loss, and the variability in the rate of vitality loss. These rates are related to body mass, environmental conditions, and xenobiotic stressors, resulting in a model that characterizes intrinsic and extrinsic factors that control a population's survival and the distribution of vitality of its individuals. The model assumes that these factors contribute to the rate parameters additively and linearly. The model is evaluated with case studies across a range of species exposed to natural and xenobiotic stressors. The mean rate of vitality loss generally is the dominant factor in determining the shape of survival curves under optimal conditions. Xenobiotic stressors add to the mean rate in proportion to the strength of the stressor. The base, or intrinsic, vitality loss rate is proportional to the −1/3 power of adult body mass across a range of iteroparous species. The increase in vitality loss rate with a xenobiotic stressor can be a function of body mass according to the allometric relationship of the organism structures affected by the stressor. The model's applicability to dose–response studies is illustrated with case studies including natural stressors (temperature, feeding interval, and population density) and xenobiotic stressors (organic and inorganic toxicants). The model provides a way to extrapolate the impact of stressors measured in one environment to another environment; by characterizing how stressors alter the vitality probability distribution, it can quantify the degree to which a stressor differentiates members of a population.

A VITALITY-BASED MODEL RELATING STRESSORS AND ENVIRONMENTAL PROPERTIES TO ORGANISM SURVIVAL

A survivorship curve is shaped by the differential survivability of the organisms within the population, and a change in a survivorship curve with a stressor reflects the differential response of the organisms to the stressor. Quantifying this linkage in a simple, rigorous way is valuable for characterizing the response of populations to stressors and ultimately for understanding the evolutionary selection of individuals exposed to stressors. To quantify this stressor–individual–population linkage with as few parameters as possible, I present a simple mechanistic model describing organism survival in terms of age-dependent and age-independent mortality rates. The age-independent rate is represented by a Poisson process. For the age-dependent rate, a concept of vitality is defined, and mortality occurs when an organism's vitality is exhausted. The loss of vitality over age is represented by a continuous Brownian-motion process, the Weiner process; vitality-related mortality occurs when the random process reaches the boundary of zero vitality. The age at which vitality-related mortality occurs is represented by the Weiner-process probability distribution for first-arrival time. The basic model has three rate parameters: the rate of accidental mortality, the mean rate of vitality loss, and the variability in the rate of vitality loss. These rates are related to body mass, environmental conditions, and xenobiotic stressors, resulting in a model that characterizes intrinsic and extrinsic factors that control a population's survival and the distribution of vitality of its individuals. The model assumes that these factors contribute to the rate parameters additively and linearly. The model is evaluated with case studies across a range of species exposed to natural and xenobiotic stressors. The mean rate of vitality loss generally is the dominant factor in determining the shape of survival curves under optimal conditions. Xenobiotic stressors add to the mean rate in proportion to the strength of the stressor. The base, or intrinsic, vitality loss rate is proportional to the −1/3 power of adult body mass across a range of iteroparous species. The increase in vitality loss rate with a xenobiotic stressor can be a function of body mass according to the allometric relationship of the organism structures affected by the stressor. The model's applicability to dose–response studies is illustrated with case studies including natural stressors (temperature, feeding interval, and population density) and xenobiotic stressors (organic and inorganic toxicants). The model provides a way to extrapolate the impact of stressors measured in one environment to another environment; by characterizing how stressors alter the vitality probability distribution, it can quantify the degree to which a stressor differentiates members of a population.

A Probability Density Function Method for Modelling Liquid Fuel Sprays

A stochastic model is proposed for modelling the turbulent dispersion of liquid fuel sprays. The approach adopted is based on the evolution equation for the joint probability density function (PDF) of the droplet properties. Turbulent dispersion is described by Langevin's equation in which a Weiner process is used to represent the stochastic force term. The method leads to plausible results when applied to a kerosene spray flame and provides a rational framework for the incorporation of liquid film break up and droplet formation processes.

Modulated Weiner process

We define and develop a novel process, related to stochastic resonance, in which a particle experiences three forces: a constant drift, a zero-mean white noise, and a time-periodic modulation. Upon reaching a threshold, the particle immediately returns to the starting point. The resulting process exhibits multiple maxima in the output power at the modulation frequency as a function of the white-noise variance, multimodal first-passage-time densities, and evidence of phase locking. Our model is an extension of the Gerstein-Mandelbrot model of neuron firing to the case of periodic stimuli, and therefore has applications in neural modeling.

On the increments of Weiner process - A look through subsequences

Let W(t) be a standard Wiener process and let βt−1 = (2a t(log(t/at)+log log t)) 1/2 where a t is a nondecreasing function of t with 0 <a t ⩽ t and t−1a t nonincreasing. Let (t k) be some increasing sequence diverging to ∞. In this paper we study the almost sure behaviour of lim sup k→∞β tk sup 0⩽s⩽a tk | W(t k+s) − W(t k) |.