Uhlenbeck Process Research Articles

Modified trajectory fitting estimators for multi‐regime threshold Ornstein–Uhlenbeck processes

The threshold Ornstein–Uhlenbeck process is a stochastic process governed by Ornstein–Uhlenbeck subprocesses with the th playing a role whenever the underlying process is in the th regime. In this paper, we investigate the parameter estimation for threshold Ornstein–Uhlenbeck processes with multiple thresholds. The classical trajectory fitting method does not apply in this context due to the significantly complex calculations. Hence, a modified trajectory fitting method is used to obtain the explicit formula of the estimators for the drift parameters based on continuous observations. The strong consistency and asymptotic normality are proven. Simulation studies illustrate the asymptotic behaviour of the trajectory fitting estimators.

Analysis of a stochastic Lotka–Volterra competitive system with infinite delays and Ornstein–Uhlenbeck process

In this paper, we construct and analyze a stochastic Lotka–Volterra competitive model with the Ornstein–Uhlenbeck process and infinite delays. First, we verify the existence and uniqueness of the global solution of the system with any initial value. Then, we investigate the pth moment boundedness, asymptotic pathwise estimation, and asymptotic behavior of the solutions of the stochastic system in turn. In addition, we develop sufficient conditions for the existence of a stationary distribution of positive solutions to the stochastic system by establishing a series of suitable Lyapunov functions. Finally, by solving the corresponding six-dimensional Fokker–Planck equation, we obtain the accurate expression of the local density function of the linear system corresponding to the stochastic system.

Biogeochemical niche conservatism relates to plant species diversification and life form evolution in a subtropical montane evergreen broad-leaved forest.

The evolutionary mechanisms underlying the biogeochemical niche conservatism in forests remain incompletely understood. Here we aimed to determine how the strengths of biogeochemical niche conservatism vary among elements and between life forms. We measured leaf concentrations of basal elements (C, N, P, K, Ca, and Mg) in a wide range of life forms in a subtropical montane evergreen broad-leaved forest. We found that differences in life forms such as evergreen/deciduous woody species and herbaceous/woody species significantly affected leaf elemental composition. The significant phylogenetic signal was present in leaf C, N, K, and Mg concentrations but absent in leaf P and Ca concentrations in all species. These contrasting strengths of biogeochemical niche conservatism were best generated by Ornstein-Uhlenbeck processes toward optima. Woody species were evolutionarily selected to show lower optimal leaf N, P, and K concentrations and higher optimal leaf C, Ca, and Mg concentrations than herbaceous species. The number of optima varied from the least in leaf C concentration to the most in leaf Ca concentration, suggesting the stronger convergent evolution of leaf Ca concentration. The positions of optima toward the tips were more selected in woody species, suggesting the more frequency of species-specific adaptations in woody species. The positions of optima were also selected at the nodes towards the species groupings from certain life forms (e.g., the group of 12 Polypodiales ferns in leaf Ca evolution and the group of three evergreen Theaceae species in leaf P evolution) that were converged to present similar leaf elemental composition. During the evolution of biogeochemical niche, strong correlations were found among leaf C, N, P, and K concentrations and between leaf Ca and Mg concentrations. In conclusion, the strengths of biogeochemical niche conservatism can vary among elements and between life forms due to the different tempo and mode of Ornstein-Uhlenbeck processes.

Bayesian fixed-domain asymptotics for covariance parameters in a Gaussian process model

Gaussian process models typically contain finite-dimensional parameters in the covariance function that need to be estimated from the data. We study the Bayesian fixed-domain asymptotics for the covariance parameters in a universal kriging model with an isotropic Matérn covariance function, which has many applications in spatial statistics. We show that when the dimension of domain is less than or equal to three, the joint posterior distribution of the microergodic parameter and the range parameter can be factored independently into the product of their marginal posteriors under fixed-domain asymptotics. The posterior of the microergodic parameter is asymptotically close in total variation distance to a normal distribution with shrinking variance, while the posterior distribution of the range parameter does not converge to any point mass distribution in general. Our theory allows an unbounded prior support for the range parameter and flexible designs of sampling points. We further study the asymptotic efficiency and convergence rates in posterior prediction for the Bayesian kriging predictor with covariance parameters randomly drawn from their posterior distribution. In the special case of one-dimensional Ornstein–Uhlenbeck process, we derive explicitly the limiting posterior of the range parameter and the posterior convergence rate for asymptotic efficiency in posterior prediction. We verify these asymptotic results in numerical experiments.

Dynamics and density function of a stochastic differential infectivity epidemic model with Ornstein–Uhlenbeck process

This paper considers a stochastic hybrid differential infectivity epidemic model with standard incidence perturbed by mean‐reverting Ornstein–Uhlenbeck process. Applying Lyapunov method, we first show existence and uniqueness of the global solution. Then sufficient conditions for persistence in the mean and exponential extinction of the infectious disease are obtained. Furthermore, by solving the corresponding Fokker–Planck equation, we derive that the global solution around the endemic equilibrium follows a unique probability density function. Finally, numerical simulations are employed to demonstrate the analytical results.

Finite-Data Error Bounds for Koopman-Based Prediction and Control

The Koopman operator has become an essential tool for data-driven approximation of dynamical (control) systems, e.g., via extended dynamic mode decomposition. Despite its popularity, convergence results and, in particular, error bounds are still scarce. In this paper, we derive probabilistic bounds for the approximation error and the prediction error depending on the number of training data points, for both ordinary and stochastic differential equations while using either ergodic trajectories or i.i.d. samples. We illustrate these bounds by means of an example with the Ornstein–Uhlenbeck process. Moreover, we extend our analysis to (stochastic) nonlinear control-affine systems. We prove error estimates for a previously proposed approach that exploits the linearity of the Koopman generator to obtain a bilinear surrogate control system and, thus, circumvents the curse of dimensionality since the system is not autonomized by augmenting the state by the control inputs. To the best of our knowledge, this is the first finite-data error analysis in the stochastic and/or control setting. Finally, we demonstrate the effectiveness of the bilinear approach by comparing it with state-of-the-art techniques showing its superiority whenever state and control are coupled.

A class of fractional Ornstein–Uhlenbeck processes mixed with a Gamma distribution

We consider a sequence of fractional Ornstein–Uhlenbeck processes, that are defined as solutions of a family of stochastic Volterra equations with a kernel given by the Riesz derivative kernel, and leading coefficients given by a sequence of independent Gamma random variables. We construct a new process by taking the empirical mean of this sequence. In our framework, the processes involved are not Markovian, hence the analysis of their asymptotic behaviour requires some ad hoc construction. In our main result, we prove the almost sure convergence in the space of trajectories of the empirical means to a given Gaussian process, which we characterize completely.

Modelling and parameter estimation for discretely observed fractional iterated Ornstein–Uhlenbeck processes

In this work we present how to model an observed time series by a FOU(p) process. We will show that the FOU(p) processes can be used to model a wide range of time series varying from short range dependence to long range dependence, with performance similar to the ARMA or ARFIMA models and in several cases outperforming them. Also, we extend the theoretical results for any FOU(p) processes for the case in which the Hurst parameter is less than 1/2 and we show theoretically and by simulations that under some conditions on T and the sample size n it is possible to obtain consistent estimators of the parameters when the process is observed in a discretized and equispaced interval [0,T]. Lastly, we give a way to obtain explicit formulas for the auto-covariance function for any FOU(p) and we present an application for FOU(2) and FOU(3).

Valuing the distant future under stochastic resettings: the effect on discounting

We investigate the effects of resetting mechanisms when valuing the future in economic terms through the discount function. Discounting is specially significant in addressing environmental problems and in evaluating the sense of urgency to act today to prevent or mitigate future losses due to climate change effects and other disasters. Poissonian resetting events can be seen in this context as a way to intervene the market, it modifies the discount function and it can facilitate a specific climate policy. We here obtain the exact expression of the discount function in Laplace space and attain the expression of the long-run interest rate, a crucial value in environmental economics and climate policy. Both quantities are obtained without assuming any model for the evolution of the market. Model specific results are achieved for diffusion processes and in particular for the Ornstein–Uhlenbeck and Feller processes. The effect of Poissonian resetting events is non-trivial in these cases. The overall lesson we can learn from the obtained results is that effective policies to favor climate action should be resolute and frequent enough in time: the frequency of the interventions is critical for actually observing the desired consequences in the long-run interest rate.

Hydrodynamics of a class of [formula omitted]-urn linear systems

In this paper we are concerned with hydrodynamics of a class of N-urn linear systems, which include voter models, pair-symmetric exclusion processes and binary contact path processes on N urns as special cases. We show that the hydrodynamic limit of our process is driven by a C[0,1)′-valued linear ordinary differential equation and the fluctuation of our process, i.e, central limit theorem from the hydrodynamic limit, is driven by a C[0,1)′-valued Ornstein–Uhlenbeck process. To derive above main results, we need several replacement lemmas. An extension in linear systems of Chapman–Kolmogorov equation plays key role in proofs of these replacement lemmas.

Non-zero-sum reinsurance and investment game with non-trivial curved strategy structure under Ornstein–Uhlenbeck process

ABSTRACT This paper investigates a non-zero-sum stochastic differential game between two competitive CARA insurers, where we adopt the different classes of premium principles (including the expected value premium principle, the variance premium principle and the exponential premium principle) and each insurer aims to maximize the expected exponential utility of his terminal wealth relative to that of his competitor. Moreover, both insurers are allowed to purchase reinsurance treaty to mitigate individual claim risks and can invest in a financial market consisting of a risk-free asset, a risky asset where the instantaneous rate of investment return follows an Ornstein-Uhlenbeck process which can reflect the changes of bull market and bear market. The optimal reinsurance strategy has a non-trivial structure which is distinguished from the conventional proportional and excess-of-loss reinsurance strategies. Furthermore, we derive the optimal reinsurance and investment strategies under the variance premium principle and expected value principle. In addition, we give another model which considers the correlation between risk model and financial market under the expected value principle. Finally, numerical analyses are provided to analyze the effects of model parameters on the optimal strategies under different cases.

Effect of Polytomy on the Parameter Estimation and Goodness of Fit of Phylogenetic Linear Regression Models for Trait Evolution

Phylogenetic regression models for trait evolution incorporate phylogenetic trees for the analysis of comparative data, in order to study trait relationships among a group of related species. However, as such trees are estimated, it is unlikely that there will be no errors when describing the relationships among species. In particular, for polytomy trees, where the relationships within a particular clade of species are more/less certainly determined (hard/soft polytomy, respectively), results of comparative analyses obtained from models based on those phylogenetic trees may also be affected. In this study, through extensive simulations, the performances of several popular Gaussian process-based regression models (Brownian motion, BM; Ornstein–Uhlenbeck process, OU; early burst, EB), as well as branch-stretching models (Pagel’s λ,δ,κ), were evaluated by assessing their fit and parameter estimation performance when soft polytomies are presented on either the root or a clade with insufficient phylogenetic information. Comparisons of the models are conducted by either assessing the accuracy of the estimator of regression and model parameters, or using a measure of fit (AIC, r2, and mean square error). It is found that, although polytomy does not significantly impact the fit and parameter estimate within a specified model, distinguishable differences and effects may be observed among trees and models. In particular, Pagel λ model and the OU model yield more accurate estimates and provide better fitting effects than the other models (BM, EB, δ, κ). While correcting phylogeny is an essential step prior to analysis, users may also consider using more appropriate models when encountering the polytomy issue.

Parameter least-squares estimation for time-inhomogeneous Ornstein–Uhlenbeck process

Abstract We address the least-squares estimation of the drift coefficient parameter θ = ( λ , A , B , ω p ) \theta=(\lambda,A,B,\omega_{p}) of a time-inhomogeneous Ornstein–Uhlenbeck process that is observed at high frequency, in which the discretized step size ℎ satisfies h → 0 h\to 0 . In this paper, under the conditions n ⁢ h → ∞ nh\to\infty and n ⁢ h 2 → 0 nh^{2}\to 0 , we prove the consistency and the asymptotic normality of the estimators. We obtain the convergence of the parameters at rate n ⁢ h \sqrt{nh} , except for ω p \omega_{p} at n 3 ⁢ h 3 \sqrt{n^{3}h^{3}} . To verify our theoretical findings, we do a simulation study. We then illustrate the use of the proposed model in fitting the energy use of light fixtures in one Belgium household and the stock exchange.

Optimal Control in the Long-Run Tracking Problem for the Exponential Ornstein–Uhlenbeck Process

Optimal Control in the Long-Run Tracking Problem for the Exponential Ornstein–Uhlenbeck Process

Influence of different kinetic heating mechanisms on the dynamics of a trapped Brownian particle in a complex plasma

Combined effect of random forces of different origins and electrostatic confinement on the dynamics of a charged Brownian particle in a plasma is investigated. Analytical equations for the effective kinetic temperature, mean square displacement (MSD), mass transfer, and velocity autocorrelation functions (VAF) of a free and trapped microparticle under the action of two random forces governed by Gaussian and Ornstein–Uhlenbeck processes are derived. The results obtained are considered on examples of solitary passive microparticles with fluctuating charges due to the discrete charging process or inhomogeneity of the surrounding plasma, active light-absorbing particles with rotational fluctuations, and strongly coupled systems with nonreciprocal interparticle interactions. The applicability of the Langevin equations with a thermal noise temperature above the ambient gas temperature to model the dynamics of dust particles in a complex plasma is discussed. It is shown that, under certain conditions, the well-known Uhlenbeck–Ornstein–Wang equations for the MSD and VAF fail to describe the oscillations of both a solitary particle and a particle in a strongly coupled monolayer system suspended in a gas discharge.

AbmR: An R package for agent‐based model analysis of large‐scale movements across taxa

Abstract Agent‐based modelling (ABM) shows promise for animal movement studies. However, a robust, open‐source and spatially explicit ABM coding platform is currently lacking. We present abmR, an R package for conducting continental‐scale ABM simulations across animal taxa. The package features two movement functions, each of which relies on the Ornstein–Uhlenbeck (OU) process. The theoretical background for abmR is discussed and the main functionalities are illustrated using example populations. Potential future additions to this open‐source package may include the ability to specify multiple environmental variables or to model interactions between agents. Additionally, updates may offer opportunities for disease ecology and integration with other R movement modelling packages.

Stability of higher order eigenvalues in dimension one

We study stability of the eigenvalues of the generator of a one dimensional reversible diffusion process satisfying some natural conditions. The proof is based on Stein’s method. In particular, these results are applied to the Normal distribution (via the Ornstein–Uhlenbeck process), to Gamma distributions (via the Laguerre process) and to Beta distributions (via the Jacobi process).

Dynamical behaviors of a stochastic HTLV-I infection model with general infection form and Ornstein–Uhlenbeck process

In this study, considering the Ornstein–Uhlenbeck process to perturb the infection rate, we develop a HTLV-I infection model with general infection form. By constructing several suitable Lyapunov functions and a compact set, and then using the strong law of numbers and Fatou’s lemma, we obtain sufficient conditions for the existence and uniqueness of the ergodic stationary distribution η(⋅) for the stochastic model. This implies long-term persistence of HTLV-I infection in a biological sense. Moreover, by using Itô’s integral stochastic model is transformed into the corresponding linearized system. Then solving the Fokker–Planck equation, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. In addition, sufficient conditions for the extinction of HTLV-I infection are established. Finally, considering different incidence rate functions, we employ numerical simulations to support our results.

Quantum memory-assisted entropic uncertainty and entanglement dynamics: two qubits coupled with local fields and Ornstein Uhlenbeck noise

Entropic uncertainty and entanglement are two distinct aspects of quantum mechanical procedures. The estimation of the entropic uncertainty is required for the increased accuracy and, hence, for the enhanced efficacy of quantum information processing tasks. In this regard, we analyze the entropic uncertainty, entropic uncertainty lower bound, and concurrence dynamics in two non-interacting qubits. The exposure of two qubits is studied in two different qubit-noise configurations, namely, common qubit-noise and independent qubit-noise interactions. To include the noisy effects of the local external fields, a Gaussian Ornstein–Uhlenbeck process is considered. We show that the increase in entropic uncertainty gives rise to the disentanglement in the two-qubit Werner state. Depending on the parameters adjustment and the number of environments coupled, different classical environments have varying capacities to induce entropic uncertainty and disentanglement in quantum systems. The entanglement remains vulnerable to the external classical fields; however, by employing the ideal parameter ranges we provided, prolonged entanglement retention while preventing entropic uncertainty growth can be achieved. Besides, we have also analyzed the intrinsic behavior of the classical fields towards two-qubit entanglement with no imperfection with respect to different parameters.

Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions

Bifurcating Markov chains (BMC) are Markov chains indexed by a full binary tree representing the evolution of a trait along a population where each individual has two children. We provide a central limit theorem for general additive functionals of BMC, and prove the existence of three regimes. This corresponds to a competition between the reproducing rate (each individual has two children) and the ergodicity rate for the evolution of the trait. This is in contrast with the work of Guyon (Ann. Appl. Probab. 17 (2007) 1538–1569), where the considered additive functionals are sums of martingale increments, and only one regime appears. Our result can be seen as a discrete time version, but with general trait evolution, of results in the time continuous setting of branching particle system from Adamczak and Miłoś (Electron. J. Probab. 20 (2015) 42), where the evolution of the trait is given by an Ornstein–Uhlenbeck process.