Ornstein-Uhlenbeck Stochastic Process Research Articles

Impact of wind power generation on a large scale power system using stochastic linear stability

The effect of random and sustained disturbances is studied in this paper. The Ornstein–Uhlenbeck stochastic process is proposed with the aim to represent the variations in the power output produced by wind power generation, considering measurements of a wind farm located in the fourth region of Chile as real field data for its parameters calibration. An example is provided considering the study of the exponential stability using the Lyapunov exponent as an indicator, with the aim to determine the maximum size of a wind farm that can be connected in a busbar of one of Chile’s interconnected power system (Sistema Interconectado del Norte Grande, SING), considering successive increments from the output power from the connected wind farm.

The nonequilibrium glassy dynamics of self-propelled particles.

We study the glassy dynamics taking place in dense assemblies of athermal active particles that are driven solely by a nonequilibrium self-propulsion mechanism. Active forces are modeled as an Ornstein-Uhlenbeck stochastic process, characterized by a persistence time and an effective temperature, and particles interact via a Lennard-Jones potential that yields well-studied glassy behavior in the Brownian limit, which is obtained as the persistence time vanishes. By increasing the persistence time, the system departs more strongly from thermal equilibrium and we provide a comprehensive numerical analysis of the structure and dynamics of the resulting active fluid. Finite persistence times profoundly affect the static structure of the fluid and give rise to nonequilibrium velocity correlations that are absent in thermal systems. Despite these nonequilibrium features, for any value of the persistence time we observe a nonequilibrium glass transition as the effective temperature is decreased. Surprisingly, increasing departure from thermal equilibrium is found to promote (rather than suppress) the glassy dynamics. Overall, our results suggest that with increasing persistence time, microscopic properties of the active fluid change quantitatively, but the general features of the nonequilibrium glassy dynamics observed with decreasing the effective temperature remain qualitatively similar to those of thermal glass-formers.

A test of financial time-series data to discriminate among lognormal, Gaussian and square-root random walks

This paper aims to offer a testing framework for the structural properties of the Brownian motion of the underlying stochastic process of a time series. In particular, the test can be applied to financial time-series data and discriminate among the lognormal random walk used in the Black-Scholes-Merton model, the Gaussian random walk used in the Ornstein-Uhlenbeck stochastic process, and the square-root random walk used in the Cox, Ingersoll and Ross process. Alpha-level hypothesis testing is provided. This testing framework is helpful for selecting the best stochastic processes for pricing contingent claims and risk management.

On the Cauchy problem for non-local Ornstein–Uhlenbeck operators

We study the Cauchy problem involving non-local Ornstein–Uhlenbeck operators in finite and infinite dimensions. We prove classical solvability without requiring that the Lévy measure corresponding to the large jumps part has a first finite moment. Moreover, we determine a core of regular functions which is invariant for the associated transition Markov semigroup. Such a core allows to characterize the marginal laws of the Ornstein–Uhlenbeck stochastic process as unique solutions to Fokker–Planck–Kolmogorov equations for measures.

Two-dimensional dynamics of expansion of a degenerate Bose gas

Expansion of a degenerate Bose gas released from a pancakelike trap is numerically simulated under the assumption of separation of the motion in the plane of the loose initial trapping and the motion in the direction of the initial tight trapping. The initial conditions for the phase fluctuations are generated using the extension to the two-dimensional case of the description of the phase noise by the Ornstein-Uhlenbeck stochastic process. The numerical simulations, taking into account both the finite size of the two-dimensional system and the atomic interactions, which cannot be neglected on the early stage of expansion, did not reproduce the scaling law for the peaks in the density fluctuation spectra experimentally observed by Choi, Seo, Kwon, and Shin [Phys. Rev. Lett. 109, 125301 (2012)]. The latter experimental results may thus require an explanation beyond our current assumptions.

Estimation of Time-Dependent Input from Neuronal Membrane Potential

The set of firing rates of the presynaptic excitatory and inhibitory neurons constitutes the input signal to the postsynaptic neuron. Estimation of the time-varying input rates from intracellularly recorded membrane potential is investigated here. For that purpose, the membrane potential dynamics must be specified. We consider the Ornstein-Uhlenbeck stochastic process, one of the most common single-neuron models, with time-dependent mean and variance. Assuming the slow variation of these two moments, it is possible to formulate the estimation problem by using a state-space model. We develop an algorithm that estimates the paths of the mean and variance of the input current by using the empirical Bayes approach. Then the input firing rates are directly available from the moments. The proposed method is applied to three simulated data examples: constant signal, sinusoidally modulated signal, and constant signal with a jump. For the constant signal, the estimation performance of the method is comparable to that of the traditionally applied maximum likelihood method. Further, the proposed method accurately estimates both continuous and discontinuous time-variable signals. In the case of the signal with a jump, which does not satisfy the assumption of slow variability, the robustness of the method is verified. It can be concluded that the method provides reliable estimates of the total input firing rates, which are not experimentally measurable.

Fluctuations and Stochastic Processes in One-Dimensional Many-Body Quantum Systems

We study the fluctuation properties of a one-dimensional many-body quantum system composed of interacting bosons and investigate the regimes where quantum noise or, respectively, thermal excitations are dominant. For the latter, we develop a semiclassical description of the fluctuation properties based on the Ornstein-Uhlenbeck stochastic process. As an illustration, we analyze the phase correlation functions and the full statistical distributions of the interference between two one-dimensional systems, either independent or tunnel-coupled, and compare with the Luttinger-liquid theory.

Longitudinal Data Analysis with Event Time as a Covariate

We consider the estimation of a nonparametric smooth function of some event time in a semiparametric mixed effects model from repeatedly measured data when the event time is subject to right censoring. The within-subject correlation is captured by both cross-sectional and time-dependent random effects, where the latter is modeled by a nonhomogeneous Ornstein–Uhlenbeck stochastic process. When the censoring probability depends on other variables in the model, which often happens in practice, the event time data are not missing completely at random. Hence, the complete case analysis by eliminating all the censored observations may yield biased estimates of the regression parameters including the smooth function of the event time, and is less efficient. To remedy, we derive the likelihood function for the observed data by modeling the event time distribution given other covariates. We propose a two-stage pseudo-likelihood approach for the estimation of model parameters by first plugging an estimator of the conditional event time distribution into the likelihood and then maximizing the resulting pseudo-likelihood function. Empirical evaluation shows that the proposed method yields negligible biases while significantly reduces the estimation variability. This research is motivated by the project of hormone profile estimation around age at the final menstrual period for the cohort of women in the Michigan Bone Health and Metabolism Study.

Simulation and inference for stochastic volatility models driven by Levy processes

SUMMARY We study Ornstein-Uhlenbeck stochastic processes driven by Levy processes, and extend them to more general non-Ornstein-Uhlenbeck models. In particular, we investigate the means of making the correlation structure in the volatility process more flexible. For one model, we implement a method for introducing quasi long-memory into the volatility model. We demonstrate that the models can be fitted to real share price returns data.

Transparent boundary conditions as dissipative subgrid closures for the spectral representation of scalar advection by shear flows

We consider the evolution of a passive scalar in a shear flow in its representation as a system of lattice differential equations in wave number space. When the velocity field has small support, the interaction in wave number space is local and can be studied in terms of dispersive linear lattice waves. We close the restriction of the system to a finite set of wave numbers by implementing transparent boundary conditions for lattice waves. This closure is studied numerically in terms of energy dissipation rate and energy spectrum, both for a time-independent velocity field and for a time-dependent synthetic velocity field whose Fourier coefficients follow independent Ornstein-Uhlenbeck stochastic processes.

An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein–Uhlenbeck stochastic process

An estimate of the probability that the queue length exceeds the maximum for a queue that is a generalized Ornstein–Uhlenbeck stochastic process

Decoherence induced by squeezing control errors in optical and ion trap holonomic quantum computations

We study decoherence induced by stochastic squeezing control errors considering the particular implementation of Hadamard gate on optical and ion trap holonomic quantum computers. We find the fidelity for Hadamard gate and compute the purity of the final state when the control noise is modeled by Ornstein-Uhlenbeck stochastic process. We demonstrate that in contradiction to the case of the systematic control errors the stochastic ones lead to decoherence of the final state. In the small errors limit we derive a simple formulae connecting the gate fidelity and the purity of the final state.

Line shapes in CP/MAS 13C NMR spectra of cellulose I

The CP/MAS 13C NMR line shape of cellulose I has been qualitatively analyzed by direct simulations using the Ornstein–Uhlenbeck stochastic process and the Kubo model. Both approaches describe a anhydroglucose C4 carbon as a oscillator with fluctuating Larmor frequency. The NMR resonance frequency is written ω = ω ¯ + ω ( t ) , where the fluctuating part with zero mean was modelled as a stationary Markov diffusion process. The simulation results both motivates the use of multiple line shapes when fitting CP/MAS 13C NMR spectra recorded on cellulose I and gives some insights into why signals from crystalline cellulose I give rise to Lorentzian line shapes.

A contango-constrained model for storable commodity prices

In this article we develop a model for the commodity price dynamics under the risk-neutral measure where the spot price switches between two distinct stochastic processes depending on whether or not inventory is being held. Speciflcally, whenever the drift of the spot price exceeds the cost of carrying inventory (interest rate plus storage costs) the inventory is being held. Conversely, whenever the drift of the spot price is less than the cost of carry, all the inventory is sold and the storage facility becomes empty. If inventory is being held, we assume that the spot price follows a geometric Brownian motion with drift equal to the cost of carrying inventory. Otherwise, the price follows a Ornstein-Uhlenbeck stochastic process. This model verifles arbitrage-free arguments since the commodity price process has a drift lees or equal to the cost of carry under the risk neutral measure. We illustrate and analyze the properties of the spot price and the forward curves implied by this model using numerical examples. The spot price sample paths and the corresponding forward curves are constructed by applying trinomial tree techniques. For comparison, we also provide the equivalent numerical examples for the single-factor model provided by Schwartz (1997), which correspond to the unconstrained version of the spot price process in this model.

Biometric random number generators

Up to now biometric methods have been used in cryptography for authentication purposes. In this paper we propose to use biological data for generating sequences of random bits. We point out that this new approach could be particularly useful to generate seeds for pseudo-random number generators and so-called “key sessions”. Our method is very simple and is based on the observation that, for typical biometric readings, the last binary digits fluctuate “randomly”. We apply our method to two data sets, the first based on animal neurophysiological brain responses and the second on human galvanic skin response. For comparison we also test our approach on numerical samplings of the Ornstein–Uhlenbeck stochastic process. To verify the randomness of the sequences generated, we apply the standard suite of statistical tests (FIPS 140-2) recommended by the National Institute of Standard and Technology for studying the quality of the physical random number generators, especially those implemented in cryptographic modules. Additionally, to confirm the high cryptographic quality of the biometric generators, we also use the often recommended Maurer's universal test and the Lempel–Ziv complexity test, which estimate the entropy of the source. The results of all these verifications show that, after appropriate choice of encoding and experimental parameters, the sequences obtained exhibit excellent statistical properties, which opens the possibility of a new design technology for true random number generators. It remains a challenge to find appropriate biological phenomena characterized by easy accessibility, fast sampling rate, high accuracy of measurement and variability of sampling rate.

-Scaling of dNch/d at by the PHOBOS Collaboration and the Ornstein-Uhlenbeck Stochastic Process

Using the latest data concerning $dN_{\rm ch}/d\eta$ at $\sqrt{s_{NN}} = 200$ GeV, we have analyzed them by means of stochastic theory named the Ornstein-Uhlenbeck process with two sources. Moreover, we display that $z_r = \eta/\sqrt{< \eta^2>} = \eta/\eta_{rms}$ scaling with centrality cuts are described by two Gaussian distributions.

Parametric dynamics of level spacings in quantum chaos

We identify parametric (radial) Bessel–Ornstein–Uhlenbeck stochastic processes as primitive dynamical models of energy level repulsion in irregular quantum systems. Familiar GOE, GUE, GSE and non-Hermitian Ginibre-type (Wigner surmise) level spacing distributions arise as special cases in that formalism, as densities of asymptotic invariant (equilibrium) probability measures.

Jointly Modeling Longitudinal and Event Time Data With Application to Acquired Immunodeficiency Syndrome

In many clinical and epidemiologic studies, periodically measured disease markers are used to monitor progression to the onset of disease. Motivated by a study of CD4 counts in men infected with human immunodeficiency virus (HIV) at risk for acquired immunodeficiency syndrome (AIDS), we developed a joint model for analysis of both longitudinal and event time data. We use a longitudinal model for continuous data that incorporates a mean structure dependent on covariates, a random intercept, a stochastic process, and measurement error. A central component of the longitudinal model is an integrated Ornstein–Uhlenbeck stochastic process, which represents a family of covariance structures with a random effects model and Brownian motion as special cases. The regression model for the event time data is a proportional hazards model that includes the longitudinal marker as a time-dependent variable and other covariates. A Markov chain Monte Carlo algorithm was developed for fitting the joint model. The joint modeling approach is evaluated and compared with the approach of separate modeling through simulation studies, and it is applied to CD4 counts and AIDS event time data from a cohort study of HIV-infected men. The joint estimation approach allows the simultaneous estimation of the effect of baseline covariates on the progression of CD4 counts and the effect of the current CD4 count and baseline covariates on the hazard of AIDS. The joint modeling approach also gives a way to incorporate measurement error in CD4 counts into a hazard model.

Does the covariance structure matter in longitudinal modelling for the prediction of future CD4 counts?

We investigate the importance of the assumed covariance structure for longitudinal modelling of CD4 counts. We examine how individual predictions of future CD4 counts are affected by the covariance structure. We consider four covariance structures: one based on an integrated Ornstein-Uhlenbeck stochastic process; one based on Brownian motion, and two derived from standard linear and quadratic random-effects models. Using data from the Multicenter AIDS Cohort Study and from a simulation study, we show that there is a noticeable deterioration in the coverage rate of confidence intervals if we assume the wrong covariance. There is also a loss in efficiency. The quadratic random-effects model is found to be the best in terms of correctly calibrated prediction intervals, but is substantially less efficient than the others. Incorrectly specifying the covariance structure as linear random effects gives too narrow prediction intervals with poor coverage rates. Fitting using the model based on the integrated Ornstein-Uhlenbeck stochastic process is the preferred one of the four considered because of its efficiency and robustness properties. We also use the difference between the future predicted and observed CD4 counts to assess an appropriate transformation of CD4 counts; a fourth root, cube root and square root all appear reasonable choices.

Lagrangian dynamics in high-dimensional point-vortex systems

We study the Lagrangian dynamics of systems of N point vortices and passive particles in a two-dimensional, doubly periodic domain. The probability distribution function of vortex velocity, pN, has a slow-velocity Gaussian component and a significant high-velocity tail caused by close vortex pairs. In the limit for N→∞, pN tends to a Gaussian. However, the form of the single-vortex velocity causes very slow convergence with N; for N≈106 the non-Gaussian high-velocity tails still play a significant role. At finite N, the Gaussian component is well modeled by an Ornstein-Uhlenbeck (OU) stochastic process with variance σN=N ln N/2π. Considering in detail the case N=100, we show that at short times the velocity autocorrelation is dominated by the Gaussian component and displays an exponential decay with a short Lagrangian decorrelation time. The close pairs have a long correlation time and cause nonergodicity over at least the time of the integration. Due to close vortex dipoles the absolute dispersion differs significantly from the OU prediction, and shows evidence of long-time anomalous dispersion. We discuss the mathematical form of a new stochastic model for the Lagrangian dynamics, consisting of an OU model combined with long-lived close same-sign vortices engaged in rapid rotation and long-lived close dipoles engaged in ballistic motion. From a dynamical-systems perspective this work indicates that systems of dimension O(100) can have behavior which is a combination of both low-dimensional behavior, i.e., close pairs, and extremely high-dimensional behavior described by traditional stochastic processes.