Ornstein-Uhlenbeck Process Research Articles (Page 1)

ABSTRACT The pneumonia caused by coronavirus infection seriously threatens the lives and health of the people and is a war without fire. The paper considers the COVID‐19 model with susceptible, latent, asymptomatic, symptomatic, and rehabilitative stages. First, we study the local stability of boundary equilibrium points for deterministic systems. Then, the model is dimensionally reduced by a limit set, and the method of Lyapunov functions is used to prove that the endemic equilibrium point is locally asymptotically stable for the dimensionally reduced model. For stochastic systems with Ornstein–Uhlenbeck processes, we first prove the existence and uniqueness of positive solution. In addition, in‐depth research was conducted on the persistence and extinction of the disease. And the density function near the positive equilibrium point is described in detail. Finally, some numerical simulations help us verify the above conclusions.

Analytical solutions for time-fractional Cauchy problem based on OU, CIR and Jacobi processes with time-dependent parameters

Analyzing HIV transmission through a stochastic system with the log-normal Ornstein–Uhlenbeck process

Dynamical impact of virus carrier screening and actively seeking treatment on a stochastic HIV/AIDS infection model with log-normal Ornstein–Uhlenbeck process

The impact of acid-base changes on the stochastic dynamics of phytoplankton growth under global warming.

Our analysis of statistical arbitrage is based on the assumption that the spread of two assets follows a mean-reversing OrnsteinUhlenbeck process when two assets are paired. By incorporating the cointegration test, Hurst component, and optimization into this framework, we develop a technique to identify the best pair in the market and verify the profitability of the pair trading strategy in Chinas A-share Market. Although individual pairs may exhibit periods of inconsistency, constructing a diversified portfolio of multiple pairs significantly enhances performance.

Jerky active particles are Brownian self-propelled particles which are dominated by “jerk,” the change in acceleration. They represent a generalization of inertial active particles. In order to describe jerky active particles, a linear jerk equation of motion which involves a third-order derivative in time, Stokes friction, and a spring force is combined with activity modeled by an active Ornstein-Uhlenbeck process. This equation of motion is solved analytically and the associated mean-square displacement (MSD) is extracted as a function of time. For small damping and small spring constants, the MSD shows an enormous superballistic spreading with different scaling regimes characterized by anomalous high dynamical exponents 6, 5, 4, or 3 arising from a competition among jerk, inertia, and activity. When exposed to a harmonic potential, the gigantic spreading tendency induced by jerk gives rise to an enormous increase of the kinetic temperature and even to a sharp localization-delocalization transition, i.e., a jerky particle can escape from harmonic confinement. The transition can be either first or second order as a function of jerkiness. Finally it is shown that self-propelled jerky particles governed by the basic equation of motion can be realized experimentally both in feedback-controlled macroscopic particles and in active colloids governed by friction with memory.

Considering the influence of quarantine and vaccination factors, this study examines an SEIQRV infectious disease model that incorporates an Ornstein-Uhlenbeck process and a general incidence function. By accounting for disease-induced mortality rates among infected individuals, the article establishes the existence and uniqueness of a global solution for any arbitrary positive initial value. An adequate condition for disease extinction is also provided. Simultaneously, by reconstructing a sequence of random Lyapunov functions, we demonstrate the existence of a unique stationary distribution indicating that the disease persists over a period of time in a biological sense. Based on these findings, the precise expression for the probability density function of the stochastic model near the quasi-equilibrium state is derived. Finally, the theoretical results are verified through a series of numerical simulations.

Estimation of Ornstein–Uhlenbeck process using ultra-high-frequency data with application to intraday pairs trading strategy

We consider overdamped Langevin diffusions in Euclidean space, with curvature equal to the spectral gap. This includes the Ornstein–Uhlenbeck process as well as non Gaussian and non product extensions with convex interaction, such as the Dyson process from random matrix theory. We show that a cutoff phenomenon or abrupt convergence to equilibrium occurs in high-dimension, at a critical time equal to the logarithm of the dimension divided by twice the spectral gap. This cutoff holds for Wasserstein distance, total variation, relative entropy, and Fisher information. A key observation is a relation to a spectral rigidity, linked to the presence of a Gaussian factor. A novelty is the extensive usage of functional inequalities, even for short-time regularization, and the reduction to Wasserstein. The proofs are short and conceptual. Since the product condition is satisfied, an Lp cutoff holds for all p. We moreover discuss a natural extension to Riemannian manifolds, a link with logarithmic gradient estimates in short-time for the heat kernel, and ask about stability by perturbation. Finally, beyond rigidity but still for diffusions, a discussion around the recent progresses on the product condition for nonnegatively curved diffusions leads us to introduce a new curvature product condition.

The paper discusses the stochastic dynamics of the vortex shedding process in the presence of external harmonic excitation and coloured multiplicative noise. The situation is encountered in a turbulent practical combustor experiencing combustion instability. Acoustic feedback and turbulent flow are imitated by the harmonic and stochastic excitations, respectively. The Ornstein–Uhlenbeck process is used to generate the noise. A low-order model for vortex shedding is used. The Fokker–Planck framework is used to obtain the evolution of the probability density function of the shedding time period. Stochastic lock-in and resonance characteristics are studied for various parameters associated with the harmonic (amplitude, frequency) and noise (amplitude, correlation time, multiplicative noise factor) excitations. We observed that: (i) the stochastic lock-in (s-lock-in) boundary strongly depends on the noise correlation time; (ii) the parameter sites for s-lock-in can be approximately identified from the noise-induced shedding statistics; and (iii) stochastic resonance is significant for some intermediate correlation times. The effects of the above-mentioned observations are discussed in the context of combustion instability.

Abstract Linear state space models provide a useful framework for investigating phenotypic evolution in fossil lineages for a wide variety of models representing Brownian motion, Ornstein-Uhlenbeck processes, and the potential influence of environmental covariates. A state space framework also provides access to residuals for the predicted and observed values at each time point as well as improved numerical stability. We illustrate the value of the state space approach by reanalyzing a classic dataset of trait evolution in the diatom lineage Stephanodiscus yellowstonensis. A series of increasingly complex models were fit to these data, including a novel modification of an Ornstein-Uhlenbeck model in which a trait tracks an exogenous covariate. These model results suggest that the number of spines on the periphery of the diatom is best explained by adaptation to changing solar insolation over time.

We present Stochastic Dynamic Mode Decomposition (SDMD), a novel data-driven framework for approximating the Koopman semigroup in stochastic dynamical systems. Unlike existing approaches, SDMD explicitly incorporates sampling time into its formulation to ensure numerical stability and precision in the presence of noise. By directly approximating the Koopman semigroup rather than its generator, SDMD avoids computationally expensive matrix exponential calculation, providing a more practically efficient pathway for analyzing stochastic dynamics. The framework also leverages neural networks for automated basis selection, minimizing manual effort while preserving computational efficiency. We establish SDMD's theoretical foundations through rigorous convergence guarantees across three critical limits in order: large data, infinitesimal sampling time, and increasing dictionary size. Numerical experiments on canonical stochastic systems including oscillatory system, mean-reverting processes, metastable system, and a neural mass model demonstrate SDMD's effectiveness in capturing the spectral properties of the Koopman semigroup, even in systems with complex random behavior.

Analysis of threshold Ornstein–Uhlenbeck process with piecewise linear drift and piecewise constant diffusion

Based on WHO’s legitimated fear that there will be an avian influenza virus strain capable of mutating and mathematical modeling methods used to describe the transmission dynamics of influenza have been widely used, but the Allee effect of population due to limited resources has not been profoundly studied. In this study, we first develop a novel deterministic SI-SIR type model with weak Allee effects to depict the spread of influenza. We prove that there is a unique global positive solution to the model with any positive initial value, then we analyze the local asymptotical stability of the possible equilibria. After that, we develop a stochastic counterpart on the basis of the deterministic model where the disease transmission rates are assumed to be affected by the Ornstein-Uhlenbeck process to depict the effects of environmental noise. By developing a stochastic Lyapunov function, we establish sufficient criteria for the existence of a stationary distribution of the stochastic model which implies that all the populations in a community will coexist for a long time. Further, we also give some conditions for the exponential extinction of the infective avian population. Meanwhile, the threshold for extinction and persistence of the infective avian population is also derived. Finally, numerical simulations are presented to illustrate the theoretical results.

This study develops a stochastic SIQR epidemic model with mean-reverting Ornstein–Uhlenbeck (OU) processes for both transmission rate β(t) and quarantine release rate k(t); this is distinct from existing non-white-noise stochastic epidemic models, most of which focus on single-parameter perturbation or only stability analysis. It synchronously embeds OU dynamics into two core epidemic parameters to capture asynchronous fluctuations between infection spread and control measures. It adopts a rare measure solution framework to derive rigorous infection extinction conditions, linking OU’s ergodicity to long-term β+(t) averages. It obtains the explicit probability density function of the four-dimensional SIQR system, filling the gap of lacking quantifiable density dynamics in prior studies. Simulations validate that R0d<1 ensures almost sure extinction, while R0e>1 leads to stable stochastic persistence.

Free-space optical (FSO) communication systems face significant challenges from atmospheric turbulence, which induces time-correlated fading and burst errors that critically affect link reliability. This paper presents a comprehensive end-to-end CCSDS O3K simulation platform with detailed atmospheric channel and pointing error modeling, enabling realistic performance evaluation. The atmospheric channel model follows ITU-R P.1622-1 recommendations and incorporates amplitude scintillation with temporal correlation using Ornstein–Uhlenbeck processes, while the pointing error model captures beam misalignment effects inherent in satellite optical links. Through extensive Monte Carlo simulations, we investigate the impact of coherence time, and interleaving depth on system performance. Results show that deeper interleaving significantly improves reliability under realistic channel conditions, providing valuable design guidance for CCSDS-compliant optical communication systems. This study does not propose new algorithms or protocols; rather, it delivers the first end-to-end CCSDS-compliant simulation framework under realistically modeled turbulence and pointing errors. Accordingly, the results offer meaningful reference value and practical benchmarks for inter-satellite optical communication research and system design.

Phylogenetic comparative methods are a major tool for evaluating macroevolutionary hypotheses. Methods based on the mean-reverting stochastic Ornstein-Uhlenbeck process allow for modeling adaptation on a phenotypic adaptive landscape that itself evolves and where fitness peaks depend on measured characteristics of the external environment and/or other organismal traits. Here we give an overview of the conceptual framework for the many implementations of these methods and discuss how we might interpret estimated parameters. We emphasize that the ability to model a changing adaptive landscape sets these methods apart from other approaches and discuss why this aspect captures long-term trait evolution more realistically. Recent multivariate extensions of these methods provide a powerful framework for testing evolutionary hypotheses but are also more complicated to use and interpret. We provide some guidance on their usage and put recent literature on the topic in biological rather than mathematical terms. We further show how these methods provide a starting point for modeling reciprocal selection (i.e., coevolution) between interacting lineages. We then briefly review some critiques of the methodologies. Finally, we provide some ideas for future developments that we think will be useful to evolutionary biologists.

Berry-Esséen bounds for the statistical estimators of an Ornstein-Uhlenbeck process driven by a general Gaussian noise

Abstract We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equation for Non-Equilibrium Reversible Irreversible Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein-Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.