Search

This work is focused on finding G‐optimal designs theoretically for kriging models with two‐dimensional inputs and separable exponential covariance structures. For design comparison, the notion of evenness of two‐dimensional grid designs is developed. The mathematical relationship between the design and the supremum of the mean squared prediction error (SMSPE) function is studied and then optimal designs are explored for both prospective and retrospective design scenarios. In the case of prospective designs, the new design is developed before the experiment is conducted and the regularly spaced grid is shown to be the G‐optimal design. Retrospective designs are constructed by adding or deleting points from an already existing design. Deterministic algorithms are developed to find the best possible retrospective designs (which minimizes the SMSPE). It is found that a more evenly spread design under the G‐optimality criterion leads to the best possible retrospective design. For all the cases of finding the optimal prospective designs and the best possible retrospective designs, both frequentist and Bayesian frameworks have been considered. The proposed methodology for finding retrospective designs is illustrated with a spatiotemporal river water quality monitoring experiment. [ABSTRACT FROM AUTHOR]

We conduct a numerical investigation to assess the effect of noise characteristics on early warning indicators (EWIs) of critical transition in Van der Pol oscillators undergoing supercritical and subcritical Hopf bifurcation. Our study primarily focuses on the effects of additive correlated noise modeled by the OU process in the stable (subthreshold) region and corresponding trends in EWIs based on signal amplitude distribution, frequency spectra, fractal, and complexity measures as the system is brought closer to bifurcation, as a function of noise color and intensity. Our results indicate that the coherence factor is a reliable indicator for the entire range of investigated noise color, while variance and decay rates of the autocorrelation function are reliable when noise correlation times are either much smaller or larger than the system time scale. Kurtosis, permutation entropy and Jensen-Shannon complexity can be effectively employed in systems where noise exhibits minimal correlation time (resembling white noise). While the Hurst exponent proves a reliable indicator in systems where noise has correlation times much larger than the time scale of the system, multi-fractal spectrum width and skewness are deemed unsuitable as EWIs. These insights enhance our understanding of the effectiveness and limitations of various EWIs in forecasting critical transitions within dynamical systems. [ABSTRACT FROM AUTHOR]

Influenza remains one of the most widespread epidemics, characterized by serious pathogenicity and high lethality, posing a significant threat to public health. This paper focuses on an influenza A infection model that includes vaccination and asymptomatic patients. The deterministic model examines the existence and local asymptotic stability of equilibria. In light of the influence of environmental disruption on the spread of disease, we develop a stochastic model in which the transmission rate follows a log-normal Ornstein–Uhlenbeck process. To demonstrate the dynamic behavior of the stochastic model, we verify the existence and uniqueness of the global positive solution. The establishment of suitable Lyapunov functions allows for the determination of sufficient conditions for the stationary distribution and extinction of the disease. Furthermore, the expression of the local density function around the quasi-endemic equilibrium is represented. Eventually, numerical simulations are conducted to support theoretical results and explore the effect of environmental noise. Our findings indicate that high noise intensity can expedite the extinction of the disease, while low noise intensity can facilitate the disease reaching a stationary distribution. This information may be valuable in developing strategies for disease prevention and control. [ABSTRACT FROM AUTHOR]

The density of a population in a natural state often fluctuates greatly, but it is not an unlimited change. Throughout the full text, we consider a stochastic predator–prey system with fear, Holling-II response function, distributed delay and mean-reverting Ornstein–Uhlenbeck process, which can better reflect the actual situation and provide theoretical basis for species research in the actual environment. And we drew some significant conclusions. The first step is to obtain the stability of equilibrium point. Then, the existence and uniqueness of positive solution for stochastic system is got and we also reach the stationary distribution of the stochastic system. In addition, we present an exact local expression for the density function of the random system near the unique positive equilibrium. Then we obtain the pth moment boundedness and the asymptotic behavior of the solution. At last, we give the extinction condition of the population system by the asymptotic behavior of the solution. In particular, we confirm the theoretical results by numerical simulation in the corresponding section. [ABSTRACT FROM AUTHOR]

As the prevalence of viral infection in body, human T‐cell leukemia virus type 1 (HTLV‐1) is receiving increasing attention. Research on the corresponding virus models is of great significance to tackle the challenges of understanding HTLV‐1 development and treatment. This paper focuses on the dynamic analysis for a stochastic model with nonlinear cytotoxic T lymphocyte (CTL) response, which is driven by Ornstein–Uhlenbeck (OU) process to model the progression of HTLV‐1 in vivo. Rich dynamic behaviors such as the extinction of infected CD4+ T cells (ITCs), stationary distribution (SD), probability density, and finite‐time stability (FTS) of the model are established to reveal the interaction of cell populations. The optimal therapeutic strategy based on the cost‐benefit viewpoint is further obtained. Finally, illustrative numerical simulations are represented to corroborate the effectiveness of treatment and the ambient perturbation's impact that strengthening the noise strength can lead to rapid virus clearance. [ABSTRACT FROM AUTHOR]

Statistical inference is a critical issue in the applications of uncertain differential equations. In this paper, such a parameter estimation problem is formulated for OrnsteinUhlenbeck process driven by Liu process from discrete observation. The contrast function is given to obtain the least squares estimators. The consistency and asymptotic distribution of two estimators are derived. Some numerical simulations and an empirical analysis on the loan interest rates of RMB under the real data are provided to verify the effectiveness of the estimation methods. [ABSTRACT FROM AUTHOR]

Cholera, as an endemic disease around the world, has imposed great harmful effects on human health. In addition, from a microscopic viewpoint, the interference of random factors exists in the process of virus replication. However, there are few theoretical studies of viral infection models with biologically reasonable stochastic effects. This paper studied a stochastic cholera model used to describe transmission dynamics in China. In this paper, we adopted a special method to simulate the effect of environmental perturbations to the system instead of using linear functions of white noise, i.e., the transmission rate of environment to human was satisfied Ornstein-Uhlenbeck processes, which is a more practical and interesting. First, it was theoretically proved that the solution to the stochastic model is unique and global, with an ergodic stationary distribution. Moreover, by solving the corresponding Fokker-Planck equation and using our developed algebraic equation theory, we obtain the exact expression of probability density function around the quasi-equilibrium of the stochastic model. Finally, several numerical simulations are provided to confirm our analytical results. [ABSTRACT FROM AUTHOR]

In this paper, we provide a unified approach to simulate diffusion bridges. The proposed method covers a wide range of processes including univariate and multivariate diffusions, and the diffusions can be either time-homogeneous or time-inhomogeneous. We provide a theoretical framework for the proposed method. In particular, using the parametrix representations we show that the approximated probability transition density function converges to that of the true diffusion, which in turn implies the convergence of the approximation. Unlike most of the methods proposed in the literature, our approach does not involve acceptance-rejection mechanics. That is, it is acceptance-rejection free. Extensive numerical examples are provided for illustration and demonstrate the accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

We propose, in this paper, a novel stochastic SIRS epidemic model to characterize the effect of uncertainty on the distribution of infectious disease, where the general incidence rate and Ornstein–Uhlenbeck process are also introduced to describe the complexity of disease transmission. First, the existence and uniqueness of the global nonnegative solution of our model is obtained, which is the basis for the discussion of the dynamical behavior of the model. And then, we derive a sufficient condition for exponential extinction of infectious diseases. Furthermore, through constructing a Lyapunov function and using Fatou's lemma, we obtain a sufficient criterion for the existence and ergodicity of a stationary distribution, which implies the persistence of the disease. In addition, the specific form of the density function of the model near the quasiendemic equilibrium is proposed by solving the corresponding Fokker–Planck equation and using some relevant algebraic equation theory. Finally, we explain the above theoretical results through some numerical simulations. [ABSTRACT FROM AUTHOR]

AbstractIn this article, we investigate the trajectory fitting estimation for the integrated Ornstein-Uhlenbeck process driven by the Lévy process based on continuous observations. The strong consistency and asymptotic distributions of the estimator are obtained by the strong law of large numbers and the inner clock property of the α-stable stochastic integral. [ABSTRACT FROM AUTHOR]

To solve the problem of temperature control in proton exchange membrane electrolytic cell (PEMEC), this paper presents a temperature control method based on multi‐experience pool probability playback and Ornstein‐Uhlenbeck noise‐twin delay depth deterministic strategy gradient. Firstly, considering the influence of water supply, anode and cathode pressure, and natural heat dissipation on temperature, a refined thermal model of PEMEC is established and transformed into a Markov model under the framework of deep reinforcement learning (DRL). Then, to solve the training instability and poor control effect of DRL caused by inertia delay of the PEMEC temperature control system, multi‐empirical pool probability playback and Ornstein‐Uhlenbeck random process noise techniques are introduced on the basis of the traditional DRL method. Finally, the simulation and hardware‐in‐the‐loop experience results show that the proposed method outperforms other advanced methods. [ABSTRACT FROM AUTHOR]

In this study, a cholera infection model with a bilinear infection rate is developed by considering the perturbation of the infection rate by the mean-reverting process. First of all, we give the existence of a globally unique positive solution for a stochastic system at an arbitrary initial value. On this basis, the sufficient condition for the model to have an ergodic stationary distribution is given by constructing proper Lyapunov functions and tight sets. This indicates in a biological sense the long-term persistence of cholera infection. Furthermore, after transforming the stochastic model to a relevant linearized system, an accurate expression for the probability density function of the stochastic model around a quasi-endemic equilibrium is derived. Subsequently, the sufficient condition to make the disease extinct is also derived. Eventually, the theoretical findings are shown by numerical simulations. Numerical simulations show the impact of regression speed and fluctuation intensity on stochastic systems. [ABSTRACT FROM AUTHOR]

In this paper, we propose an estimator for the Ornstein–Uhlenbeck parameters based on observations of its supremum. We derive an analytic expression for the supremum density. Making use of the pseudo-likelihood method based on the supremum density, our estimator is constructed as the maximal argument of this function. Using weak-dependency results, we prove some statistical properties on the estimator such as consistency and asymptotic normality. Finally, we apply our estimator to simulated and real data. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic Leslie-Gower model with a Holling-II functional response that is driven by the Ornstein-Uhlenbeck process. Some asymptotic properties of the solution of the stochastic Leslie-Gower model are introduced: The existence and uniqueness of the global solution of the model are given; the ultimate boundedness of the model is proven; by constructing the Lyapunov function and applying Ito’s formula, the existence of the stationary distribution of the model is demonstrated; and the conditions for system extinction are discussed. Finally, numerical simulations are used to validate our conclusion. [ABSTRACT FROM AUTHOR]

We propose a general structural model for valuing risky corporate debt securities within a two-dimensional framework. The state variables in our model include the firm's asset value, described as a geometric Brownian motion stochastic process, and the short-term interest rate, following a mean-reverting Ornstein–Uhlenbeck stochastic process. Our model accommodates flexible debt structure, multiple seniority classes, tax benefits, bankruptcy costs, and a stochastic endogenous default barrier. The proposed methodology relies on a two-dimensional dynamic program coupled with finite elements where key transition parameters are computed in closed form, and effective approximations using local interpolations are made during backward recursion. Our design incorporates space discretization without imposing time discretization, which is advantageous, particularly in the valuation of corporate bonds where exercise opportunities are often distant. Our methodology distinguishes itself by assuming a numerical error, setting it apart from statistical methods. Together, the above features establish dynamic programming coupled with finite elements as a competitive valuation approach as compared to its counterparts in the existing literature. We use parallel computing to enhance the efficiency of our methodology. We conduct a numerical and and an empirical investigation, both of which show consistency with several empirical evidence documented in the literature. [ABSTRACT FROM AUTHOR]

A new goodness-of-fit (GoF) test is proposed and investigated for the Gaussianity of the observed functional data. The test statistic is the Cramér-von Mises distance between the observed empirical characteristic functional (CF) and the theoretical CF corresponding to the null hypothesis stating that the functional observations (process paths) were generated from a specific parametric family of Gaussian processes, possibly with unknown parameters. The asymptotic null distribution of the proposed test statistic is derived also in the presence of these nuisance parameters, the consistency of the classical parametric bootstrap is established, and some particular choices of the necessary tuning parameters are discussed. The empirical level and power are investigated in a simulation study involving GoF tests of an Ornstein–Uhlenbeck process, Vašíček model, or a (fractional) Brownian motion, both with and without nuisance parameters, with suitable Gaussian and non-Gaussian alternatives. [ABSTRACT FROM AUTHOR]

In this paper, we study an SICA model with a standard incidence rate, where the contact rate β is controlled by the Ornstein–Uhlenbeck process. We first prove the existence and uniqueness of the global positive solution, and by constructing an appropriate Lyapunov function, we demonstrate that when R 0 s > 1 , the system has a stationary distribution. Furthermore, we obtain a concrete expression of the probability density function near the quasi-positive equilibrium point. By constructing another suitable Lyapunov function, we also derive a threshold value R 0 e for disease extinction, and when R 0 e < 1 , the disease extinguishes at an exponential rate. Finally, our conclusions are verified through numerical simulations. [ABSTRACT FROM AUTHOR]

We introduce Langevin sampling algorithms to field-theoretic simulations (FTSs) of polymers that, for the same accuracy, are ∼10× more efficient than a previously used Brownian dynamics algorithm that used predictor corrector for such simulations, over 10× more efficient than the smart Monte Carlo (SMC) algorithm, and typically over 1000× more efficient than a simple Monte Carlo (MC) algorithm. These algorithms are known as the Leimkuhler–Matthews (the BAOAB-limited) method and the BAOAB method. Furthermore, the FTS allows for an improved MC algorithm based on the Ornstein–Uhlenbeck process (OU MC), which is 2× more efficient than SMC. The system-size dependence of the efficiency for the sampling algorithms is presented, and it is shown that the aforementioned MC algorithms do not scale well with system sizes. Hence, for larger sizes, the efficiency difference between the Langevin and MC algorithms is even greater, although, for SMC and OU MC, the scaling is less unfavorable than for the simple MC. [ABSTRACT FROM AUTHOR]

AIDS is a chronic infectious disease that has been having a major impact on human health and social stability. In this paper, based on the deterministic SIATR model, a stochastic SIATR model is developed to account for the spread of AIDS by introducing the Ornstein–Uhlenbeck process. First, the existence and uniqueness of the global positive solution of the model are investigated. Second, a threshold R0E is set: when R0E < 1, the disease will become extinct; when R0E > 1, the disease will persist. Then, it is shown that there exists a stationary distribution for the model when R0E > 1. On this basis, we derive the exact expression for the probability density function of the model in the neighborhood of the quasi-equilibrium state. Finally, the results of the previous proof are verified by several numerical simulations. [ABSTRACT FROM AUTHOR]

The purpose of this work is to investigate a novel stochastic SIHR epidemic model, which includes a general incidence rate and mean-reversion Ornstein–Uhlenbeck process. Firstly, the existence of global positivity of the solution is testified by Lyapunov function. Secondly, this disease will be eradicated if the reproduction number R 0 s < 1 . Otherwise, if the reproduction number R 0 ∗ > 1 , then the system has a stationary distribution, which means that the pandemic will persist. In addition, an explicit expression of the probability density function for a linear system near quasi-endemic equilibrium is obtained under certain conditions. Finally, a series of numerical simulations is carried out to validate the theoretical conclusions. [ABSTRACT FROM AUTHOR]

The Holton-Lindzen-Plumb (HLP) model of the quasi-biennial oscillation (QBO) is investigated in order to assess the impact of introducing intermittency in the wave forcing. Intermittency is introduced to HLP by allowing the amplitude of the waves which force the QBO to evolve according to a stationary random process, driven by a stochastic differential equation (SDE) with an associated time scale τ. Provided that τ is much shorter than the QBO period, it is shown that the impact on the QBO of the intermittent forcing is captured by a single intermittency parameter λ, and the value of λ is proportional to τ and otherwise depends upon the details of the SDE. Numerical simulations, using a family of mean-reverting Ornstein-Uhlenbeck processes as the choice of SDE, show that the effect of increasing the intermittency parameter is invariably to decrease the QBO amplitude and increase its period. Changes to the QBO amplitude and period are indeed found to collapse onto a single curve controlled by λ, as predicted by the theory, provided that τ is small enough for the approximations used to be valid. The extension to broadband forcing is discussed in the context of stochastic gravity wave parameterisation, with the eventual goal of developing a representation of source intermittency in the most general situation with close fidelity to the physics. [ABSTRACT FROM AUTHOR]

Controlling infectious diseases has become an increasingly complex issue, and vaccination has become a common preventive measure to reduce infection rates. It has been thought that vaccination protects the population. However, there is no fully effective vaccine. This is based on the fact that it has long been assumed that the immune system produces corresponding antibodies after vaccination, but usually does not achieve the level of complete protection for undergoing environmental fluctuations. In this paper, we investigate a stochastic SIS epidemic model with incomplete inoculation, which is perturbed by the Ornstein–Uhlenbeck process and Brownian motion. We determine the existence of a unique global solution for the stochastic SIS epidemic model and derive control conditions for the extinction. By constructing two suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. Furthermore, we obtain the exact expression of the probability density function near the pseudo-equilibrium point of the stochastic model while addressing the four-dimensional Fokker–Planck equation under the same conditions. Finally, we conduct several numerical simulations to validate the theoretical results. [ABSTRACT FROM AUTHOR]

A stochastic influenza epidemic model where influenza virus can mutate into a mutant influenza virus is established to study the influence of environmental disturbance. And the transmission rate of the model is assumed to satisfy log-normal Ornstein–Uhlenbeck process. We verify that there exists a unique global positive solution to the stochastic model. By constructing proper Lyapunov functions, sufficient conditions under which the stationary distribution exists are obtained. In addition, we discuss the extinction of the disease. Furthermore, we get the accurate expression of probability density function near the endemic equilibrium of the stochastic model. Finally, several numerical simulations are carried out to verify theoretical results and examine the influence of environmental noise. [ABSTRACT FROM AUTHOR]

In this paper, we propose a stochastic virus infection model with nonlytic immune response, where the transmission rate is realistically modeled as being subject to continuous fluctuations, represented by the Ornstein–Uhlenbeck process. Firstly, we establish the existence and uniqueness of the global solution for the stochastic model and its invariant set, ensuring the robustness and applicability of model. Next, by constructing appropriate Lyapunov functions, we derive sufficient conditions for virus extinction and the existence of a stationary distribution for the stochastic model. These conditions elucidate the key dynamic behaviors, such as extinction and persistence, within the stochastic framework. [ABSTRACT FROM AUTHOR]

In this paper, we have studied stochastic differential equations with unbounded delay in fractional power spaces perturbed by fractional Ornstein–Uhlenbeck process Y H , ξ ⁢ (t) with H ∈ (1 2 , 1) . Subsequently, the existence and uniqueness of mild solution of the considered equation have been proved with fixed-point theorem. Finally, we obtain the global attracting set of the considered equations by some stochastic analysis and inequality technique. [ABSTRACT FROM AUTHOR]

The nonlinear dependence of the mean-squared displacement (MSD) on time is a common characteristic of particle transport in complex environments. Frequently, this anomalous behavior only occurs transiently before the particle reaches a terminal Fickian diffusion. This study shows that a system of hierarchically coupled Ornstein–Uhlenbeck equations is able to describe both transient subdiffusion and transient superdiffusion dynamics, as well as their sequential combinations. To validate the model, five distinct experimental, molecular dynamics simulation, and theoretical studies are successfully described by the model. The comparison includes the transport of particles in random optical fields, supercooled liquids, bedrock, soft colloidal suspensions, and phonons in solids. The model's broad applicability makes it a convenient tool for interpreting the MSD profiles of particles exhibiting transient anomalous diffusion. [ABSTRACT FROM AUTHOR]

In this study, we focus on the collective dynamics of polar active particles navigating across three distinct surfaces, each characterized by its own unique blend of topological and geometrical properties. The behavior of these active particles is influenced by a multitude of factors, including self-propulsion, inter-particle interactions, surface constraints, and under-damped stochastic forces simulated via Ornstein–Uhlenbeck processes. Our exploration unveils the prevailing collective patterns observed within these systems across three surface types: a sphere, a torus, and a landscape featuring hills and valleys, each distinguished by its specific topological and geometrical attributes. We underscore the profound impact of surface curvature and symmetry on the sustainable spatial-temporal dynamics witnessed. Our findings illuminate how the interplay between substantial surface curvature and particular symmetrical characteristics gives rise to a diverse spectrum of spatial-temporal patterns. Notably, we discern that high curvature tends to drive collective motion toward cyclic rotation on spheres and tori, or spatial-temporal periodic traveling ring patterns on landscapes with hills and valleys. Additionally, we observe that rough surfaces and the incorporation of excluded volume effects can disrupt the complexity of these collective spatial-temporal patterns. Through this investigation, we provide invaluable insight into the intricate interplay of curvature and symmetry, profoundly shaping collective behaviors among active particles across varied surfaces. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic non-autonomous chemostat model with mean-reverting Ornstein–Uhlenbeck process is constructed and analyzed, and we consider the stochastic perturbation on the washout rate. We first study the existence of global unique positive solution and stochastically ultimate boundedness for new stochastic chemostat system. Secondly, we obtain that the sufficient conditions for extinction exponentially and stochastic permanence of microorganism. Finally, we also give some numerical simulations, which show that the mean-reverting Ornstein–Uhlenbeck process is a very effective and reasonable method to introduce environmental noise into the chemostat model, and we also find that the reversion speed and volatility intensity have an important influence on the dynamical behavior of microorganism. [ABSTRACT FROM AUTHOR]

This paper deals with a class of optimal control problems which arises in advertising models with Volterra Ornstein-Uhlenbeck process representing the product goodwill. Such choice of the model can be regarded as a stochastic modification of the classical Nerlove-Arrow model that allows to incorporate both presence of uncertainty and empirically observed memory effects such as carryover or distributed forgetting. We present an approach to solve such optimal control problems based on an infinite dimensional lift which allows us to recover Markov properties by formulating an optimization problem equivalent to the original one in a Hilbert space. Such technique, however, requires the Volterra kernel from the forward equation to have a representation of a particular form that may be challenging to obtain in practice. We overcome this issue for Hölder continuous kernels by approximating them with Bernstein polynomials, which turn out to enjoy a simple representation of the required type. Then we solve the optimal control problem for the forward process with approximated kernel instead of the original one and study convergence. The approach is illustrated with simulations. [ABSTRACT FROM AUTHOR]

In this paper, a stochastic SIS epidemic infectious diseases model with double stochastic perturbations is proposed. First, the existence and uniqueness of the positive global solution of the model are proved. Second, the controlling conditions for the extinction and persistence of the disease are obtained. Besides, the effects of the intensity of volatility ξ 1 and the speed of reversion 1 on the dynamical behaviors of the model are discussed. Finally, some numerical examples are given to support the theoretical results. The results show that if the basic reproduction number ℛ 0 s < 1 , the disease will be extinct, that is to say that we can control the threshold ℛ 0 s to suppress the disease outbreak. [ABSTRACT FROM AUTHOR]

Based on the Girsanov theorem for a kind of Volterra-Gaussian process, which are the generalization of fractional Brownian motion, Liouville fractional Brownian motion, and fractional Ornstein-Uhlenbeck process, we establish the Harnack inequalities for a class of stochastic functional differential equations driven by a kind of Volterra-Gaussian processes with a subordinator by an approximation technique. Some known results have been generalized and improved. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic properties for the drift parameter estimators in the fractional Ornstein-Uhlenbeck process with periodic mean function and long range dependence. The Cremér-type moderate deviations, as well as the moderation deviation principle with explicit rate function can be obtained. [ABSTRACT FROM AUTHOR]

Online portfolio selection focuses on maximizing cumulative wealth and outputs a portfolio in each period. Anticor is a state-of-the-art algorithm in this area, but the similarity calculation method between long-period time series of different assets in the Anticor algorithm cannot effectively reflect the correlation of long-period time series with different stocks. The new algorithm(Anticor-DTW) proposed in this paper improves the original similarity distance calculation method of the algorithm by introducing dynamic time warping(DTW), which can identify similar shapes and spatial differences between different series by aligning the shortest paths. We not only simulated Anticor-DTW and Anticor on four classic stock datasets, NYSE(N), NYSE(O), TSE, and MSCI but also conducted experiments on new untested stock datasets HuShen300 and NASDAQ. All experiments indicated that Anticor-DTW outperforms Anticor. Moreover, we conducted a transaction costs experiment with exponential Ornstein-Uhlenbeck process, and the result also proved the great practicability of the Anticor-DTW algorithm in the real asset market. [ABSTRACT FROM AUTHOR]

This paper studies a stochastic two-species Schoener's competitive model with Lévy jumps by the mean-reverting Ornstein-Uhlenbeck process. First, the biological implication of introducing the Ornstein-Uhlenbeck process is illustrated. After that, we show the existence and uniqueness of the global solution. Moment estimates for the global solution of the stochastic model are then given. Moreover, by constructing the Lyapunov function and applying Itô's formula and Chebyshev's inequality, it is found that the model is stochastic and ultimately bounded. In addition, we give sufficient conditions for the extinction of species. Finally, numerical simulations are employed to demonstrate the analytical results. [ABSTRACT FROM AUTHOR]

Considering the transmission characteristics of COVID-19, we formulate a Susceptible-Exposed-Quarantine-Infected-Recovered epidemic model by five first-order differential equations to study the dynamical behaviors of diseases that have a latent period, quarantine strategy, governmental intervention and general incidence rate. After giving the basic reproduction number R 0 , conditions for the existence of equilibria and their local asymptotic stability are both investigated. However, environmental perturbations always have influence on the epidemic in the natural world. With the assumption that the transmission rate is driven by the log-normal Ornstein–Uhlenbeck process, we construct a corresponding stochastic epidemic model that incorporates environmental impacts. Based on the proof of existence and uniqueness of the global positive solution, two critical values R 0 e and R 0 s are established that can determine the extinction and persistence of disease, which are completely constituted by the basic reproduction number and random factors. By solving a changing four-dimensional Fokker–Planck equation, we calculate the exact analytical expression of the probability density function of stationary distribution near the quasi-endemic equilibrium. Finally, some numerical simulations are performed to support obtained theoretical results, and we show the sensitivity index to study the impact of each parameter on disease transmission. [ABSTRACT FROM AUTHOR]

We introduce a novel stochastic control model for the problem of a service firm interacting over time with one of its customers who probabilistically churns depending on the customer's satisfaction. The firm has two service modes available, and they determine the drift and volatility of the Brownian reward process. The firm's objective is to maximize the rewards generated over the customer's lifetime. Meanwhile, the customer might churn probabilistically if the customer's satisfaction, modeled as an Orstein–Uhlenbeck process controlled by the firm's service mode, is below a certain threshold. We build upon Markov processes with spatial delay to solve this problem, and we explicitly characterize the firm's optimal policy, which is either myopic or a sandwich policy. A sandwich policy is one in which the firm deploys the service mode with inferior reward rate when the customer satisfaction level is in a specific interval near the satisfaction threshold and uses the myopically optimal service mode for all other satisfaction levels. Specifically, we find that the firm should use the safe service mode when the customer is marginally satisfied and the risky service mode when the customer is marginally unsatisfied. We find numerically that the customer lifetime value under the optimal policy is large relative to that under the myopic policy. Our results are robust to a variety of alternative model specifications. Funding: J. Lu gratefully acknowledges financial support from Natural Science Foundation of China (NSFC) [Project 72192805]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2021.0179. [ABSTRACT FROM AUTHOR]

Consider an Ornstein–Uhlenbeck process that initially reverts to zero at a known mean-reversion rate β0, and then after some random/unobservable time, this mean-reversion rate is changed to β1. Assuming that the process is observed in real time, the problem is to detect when exactly this change occurs as accurately as possible. We solve this problem in the most uncertain scenario when the random/unobservable time is (i) exponentially distributed and (ii) independent from the process prior to the change of its mean-reversion rate. The solution is expressed in terms of a stopping time that minimises the probability of a false early detection and the expected delay of a missed late detection. Allowing for both positive and negative values of β0 and β1 (including zero), the problem and its solution embed many intuitive and practically interesting cases. For example, the detection of a mean-reverting process changing to a simple Brownian motion (β0>0 and β1=0) and vice versa (β0=0 and β1>0) finds a natural application to pairs trading in finance. The formulation also allows for the detection of a transient process becoming recurrent (β0<0 and β1≥0) as well as a recurrent process becoming transient (β0≥0 and β1<0). The resulting optimal stopping problem is inherently two-dimensional (because of a state-dependent signal-to-noise ratio), and various properties of its solution are established. In particular, we find the somewhat surprising fact that the optimal stopping boundary is an increasing function of the modulus of the observed process for all values of β0 and β1. [ABSTRACT FROM AUTHOR]

Vaccination is an important tool in disease control to suppress disease, and vaccine-influenced diseases no longer conform to the general pattern of transmission. In this paper, by assuming that the infection rate is affected by the Ornstein–Uhlenbeck process, we obtained a stochastic SIRV model. First, we prove the existence and uniqueness of the global positive solution. Sufficient conditions for the extinction and persistence of the disease are then obtained. Next, by creating an appropriate Lyapunov function, the existence of the stationary distribution for the model is proved. Further, the explicit expression for the probability density function of the model around the quasi-equilibrium point is obtained. Finally, the analytical outcomes are examined by numerical simulations. [ABSTRACT FROM AUTHOR]

Velocity gradient tensor, $A_{ij}\equiv \partial u_i/\partial x_j$ , in a turbulence flow field is modelled by separating the treatment of intermittent magnitude ($A = \sqrt {A_{ij}A_{ij}}$) from that of the more universal normalised velocity gradient tensor, $b_{ij} \equiv A_{ij}/A$. The boundedness and compactness of the $b_{ij}$ -space along with its universal dynamics allow for the development of models that are reasonably insensitive to Reynolds number. The near-lognormality of the magnitude $A$ is then exploited to derive a model based on a modified Ornstein–Uhlenbeck process. These models are developed using data-driven strategies employing high-fidelity forced isotropic turbulence data sets. A posteriori model results agree well with direct numerical simulation data over a wide range of velocity-gradient features and Reynolds numbers. [ABSTRACT FROM AUTHOR]

In this paper, a typical motion-based modelling and tracking issue is investigated for vehicle targets in linear road segment by a variable structure multiple model (VSMM) method. Vehicle target motions in a road with multiple lanes are described by typical lane keeping and lane changing manoeuvres. To describe trajectories of typical manoeuvres, an Ornstein–Uhlenbeck process and a sine half-cycle are used to model lane keeping and lane changing motions in lateral direction, respectively. Note that starting point and length of lane changing motion are unknown in target tracking. Time-varying model sets are designed based on the current motion model with different motion parameters. A VSMM tracking frame is constructed to obtain target state estimates with time-varying model set. The effectiveness of the proposed typical motion-based tracking scheme is displayed by simulation results on road vehicle tracking. [ABSTRACT FROM AUTHOR]

Controlling the time evolution of a probability distribution that describes the dynamics of a given complex system is a challenging problem. Achieving success in this endeavour will benefit multiple practical scenarios, e.g., controlling mesoscopic systems. Here, we propose a control approach blending the model predictive control technique with insights from information geometry theory. Focusing on linear Langevin systems, we use model predictive control online optimisation capabilities to determine the system inputs that minimise deviations from the geodesic of the information length over time, ensuring dynamics with minimum "geometric information variability". We validate our methodology through numerical experimentation on the Ornstein–Uhlenbeck process and Kramers equation, demonstrating its feasibility. Furthermore, in the context of the Ornstein–Uhlenbeck process, we analyse the impact on the entropy production and entropy rate, providing a physical understanding of the effects of minimum information variability control. [ABSTRACT FROM AUTHOR]

We develop a new continuous‐time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm continuously updates the SDE model's parameters using an estimate for the gradient of the stationary distribution. The gradient estimate is simultaneously updated using forward propagation of the SDE state derivatives, asymptotically converging to the direction of steepest descent. We rigorously prove convergence of the online forward propagation algorithm for linear SDE models (i.e., the multidimensional Ornstein–Uhlenbeck process) and present its numerical results for nonlinear examples. The proof requires analysis of the fluctuations of the parameter evolution around the direction of steepest descent. Bounds on the fluctuations are challenging to obtain due to the online nature of the algorithm (e.g., the stationary distribution will continuously change as the parameters change). We prove bounds for the solutions of a new class of Poisson partial differential equations (PDEs), which are then used to analyze the parameter fluctuations in the algorithm. Our algorithm is applicable to a range of mathematical finance applications involving statistical calibration of SDE models and stochastic optimal control for long time horizons where ergodicity of the data and stochastic process is a suitable modeling framework. Numerical examples explore these potential applications, including learning a neural network control for high‐dimensional optimal control of SDEs and training stochastic point process models of limit order book events. [ABSTRACT FROM AUTHOR]

This article aims to analyze a stochastic epidemic model SEIuIrR (Susceptible-exposed-undetected infected-detected infected (reported -recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein–Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.

By using the Ornstein-Uhlenbeck (OU) process to simulate random disturbances in the environment, and considering the influence of jump noise, a stochastic Gilpin-Ayala mutualism model driven by mean-reverting OU process with Lévy jumps was established, and the asymptotic behaviors of the stochastic Gilpin-Ayala mutualism model were studied. First, the existence of the global solution of the stochastic Gilpin-Ayala mutualism model is proved by the appropriate Lyapunov function. Second, the moment boundedness of the solution of the stochastic Gilpin-Ayala mutualism model is discussed. Third, the existence of the stationary distribution of the solution of the stochastic Gilpin-Ayala mutualism model is obtained. Finally, the extinction of the stochastic Gilpin-Ayala mutualism model is proved. The theoretical results were verified by numerical simulations.