Ornstein-Uhlenbeck Process Research Articles

Bond Pricing Under Sticky OU Process

Bond Pricing Under Sticky OU Process

Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns.

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

Analytical Survival Analysis of the Non-autonomous Ornstein–Uhlenbeck Process

The survival probability for a periodic non-autonomous Ornstein–Uhlenbeck process is calculated analytically using two different methods. The first uses an asymptotic approach. We treat the associated Kolmogorov Backward Equation with an absorbing boundary by dividing the domain into an interior region, centered around the origin, and a “boundary layer” near the absorbing boundary. In each region we determine the leading-order analytical solutions, and construct a uniformly valid solution over the entire domain using asymptotic matching. In the second method we examine the integral relationship between the probability density function and the mean first passage time probability density function. These allow us to determine approximate analytical forms for the exit rate. The validity of the solutions derived from both methods is assessed numerically, and we find the asymptotic method to be superior.

Long time behavior of a rumor model with Ornstein-Uhlenbeck process

In order to study the propagation of rumors under the influence of media, this paper analyzes a random rumor propagation system with Ornstein-Uhlenbeck process. By constructing the Lyapunov function, we get that the established model has a stationary distribution, which means that rumors will persist under the side effects of the media. In addition, we solve the corresponding matrix and get the exact expression of the probability density near the positive equilibrium. At the end of this paper, numerical simulations verify our results.

System response time and optimal path

Abstract In this work we study, in general terms, the optimal path for an out-of-equilibrium process from a geometric perspective constructed from a probability density function belonging to the exponential family (PDFE). With this objective we have constructed the speed of entropy production for a system out of equilibrium. This speed of entropy production is not determined with respect to the clock time t but rather with respect to a dimensionless time ξ (t) associated with a characteristic response time of each system λ (t). In this context, we demonstrate that the constant entropy production speed condition leads to the system moving on a geodesic of the manifold constructed from the probability density function. In the last part of this work we apply our general results to study the Ornstein–Uhlenbeck process (OU), which is a well-known stochastic process that strictly speaking describes the velocity of a Brownian particle under the influence of friction. As the system evolves we find that dimensionless time ξ (t) passes slower than clock time t.

Fractional Brownian motion in confining potentials: non-equilibrium distribution tails and optimal fluctuations

Abstract At long times, a fractional Brownian particle in a confining external potential reaches a non-equilibrium (non-Boltzmann)
steady state. Here we consider scale-invariant power-law potentials $V(x)\sim |x|^m$, where $m>0$, and employ the optimal fluctuation method (OFM) to determine the large-$|x|$ tails of the steady-state probability distribution $\mathcal{P}(x)$ of the particle position. The calculations involve finding the optimal (that is, the most likely) path of the particle, which determines these tails, via a minimization of the exact action functional for this system, which has recently become available. Exploiting dynamical scale invariance of the model in conjunction with the OFM ansatz, we establish the large-$|x|$ tails of $\ln \mathcal{P}(x)$ up to a dimensionless factor $\alpha(H,m)$, where $0<H<1$ is the Hurst exponent. We determine $\alpha(H,m)$ analytically (i) in the limits of $H\to 0$ and $H\to 1$, and (ii) for $m=2$ and arbitrary $H$, corresponding to the fractional Ornstein-Uhlenbeck (fOU) process. Our results for the fOU process are in agreement with the previously known exact $\mathcal{P}(x)$ and autocovariance. The form of the tails of $\mathcal{P}(x)$ yields exact conditions, in terms of $H$ and $m$, for the particle confinement in the potential.
For $H\neq 1/2$, the tails encode the non-equilibrium character of the steady state distribution, and we observe violation of time reversibility of the system except for $m=2$. To compute the optimal paths and the factor $\alpha(H,m)$ for arbitrary permissible $H$ and $m$, one needs to solve an (in general nonlinear) integro-differential equation. To this end we develop a specialized numerical iteration algorithm which
accounts analytically for an intrinsic cusp singularity of the optimal paths for $H<1/2$.


Parameter Estimation of Uncertain Differential Equations Driven by Threshold Ornstein–Uhlenbeck Process with Application to U.S. Treasury Rate Analysis

Uncertain differential equations, as an alternative to stochastic differential equations, have proved to be extremely powerful across various fields, especially in finance theory. The issue of parameter estimation for uncertain differential equations is the key step in mathematical modeling and simulation, which is very difficult, especially when the corresponding terms are driven by some complicated uncertain processes. In this paper, we propose the uncertainty counterpart of the threshold Ornstein–Uhlenbeck process in probability, named the uncertain threshold Ornstein–Uhlenbeck process, filling the gaps of the corresponding research in uncertainty theory. We then explore the parameter estimation problem under different scenarios, including cases where certain parameters are known in advance while others remain unknown. Numerical examples are provided to illustrate our method proposed. We also apply the method to study the term structure of the U.S. Treasury rates over a specific period, which can be modeled by the uncertain threshold Ornstein–Uhlenbeck process mentioned in this paper. The paper concludes with brief remarks and possible future directions.

A Stochastic Model for Transmission Dynamics of AIDS with Protection Consciousness and Log-normal Ornstein–Uhlenbeck Process

A Stochastic Model for Transmission Dynamics of AIDS with Protection Consciousness and Log-normal Ornstein–Uhlenbeck Process

Parameter estimation from an Ornstein-Uhlenbeck process with measurement noise

Parameter estimation from an Ornstein-Uhlenbeck process with measurement noise

Boundary crossing problems and functional transformations for Ornstein–Uhlenbeck processes

ABSTRACT We are interested in the law of the first passage time of an Ornstein–Uhlenbeck process to time-varying thresholds. We show that this problem is connected to the laws of the first passage time of the process to members of a two-parameter family of functional transformations of a time-varying boundary. For specific values of the parameters, these transformations appear in a realisation of a standard Ornstein–Uhlenbeck bridge. We provide three different proofs of this connection. The first is based on a similar result for Brownian motion, the second uses a generalisation of the so-called Gauss–Markov processes, and the third relies on the Lie group symmetry method. We investigate the properties of these transformations and study the algebraic and analytical properties of an involution operator which is used in constructing them. We also show that these transformations map the space of solutions of Sturm–Liouville equations into the space of solutions of the associated nonlinear ordinary differential equations. Lastly, we interpret our results through the method of images and give new examples of curves with explicit first passage time densities.

Capacity and delay performance analysis for large-scale UAV-enabled wireless networks

In this paper, we analyze the capacity and delay performance of a large-scale Unmanned Aerial Vehicle (UAV)-enabled wireless network, in which untethered and tethered UAVs deployed with content files move along with mobile Ground Users (GUs) to satisfy their coverage and content delivery requests. We consider the case where the untethered UAVs are of limited storage, while the tethered UAVs serve as the cloud when the GUs cannot obtain the required files from the untethered UAVs. We adopt the Ornstein-Uhlenbeck (OU) process to capture the mobility pattern of the UAVs moving along the GUs and derive the compact expressions of the coverage probability and capacity of our considered network. Using tools from martingale theory, we model the traffic at UAVs as a two-tier queueing system. Based on this modeling, we further derive the analytical expressions of the network backlog and delay bounds. Through numerical results, we verify the correctness of our analysis and demonstrate how the capacity and delay performance can be significantly improved by optimizing the percentage of the untethered UAVs with cached contents.

Mapping Risk–Return Linkages and Volatility Spillover in BRICS Stock Markets through the Lens of Linear and Non-Linear GARCH Models

This paper explores the influence of the risk–return relationship and volatility spillover on stock market returns of emerging economies, with a particular focus on the BRICS countries. This research is undertaken in a context where discussions on de-dollarization and the expansion of BRICS membership are gaining momentum, making it a novel and distinct exercise compared to prior studies. Utilizing econometric techniques to investigate daily market returns from 1 April 2008 to 31 March 2023, a period that witnessed major events like the global financial crisis, the COVID-19 pandemic, and the Russia–Ukraine conflict, linear and non-linear models like ARCH, GARCH, GARCH-M, EGARCH, and TGARCH, are employed to assess stock return volatility behaviour, assuming a Gaussian distribution of error terms. The diagnostic test confirms that the distribution is non-normal, stationary, and heteroscedastic. The key findings indicate a lack of the risk–return relationship across all BRICS stock markets, except for South Africa; a more pronounced effect of unpleasant news over pleasant news; a slow mean-reverting process in volatility; the EGARCH model is the best fit model as evidenced by a higher log likelihood and lower Akaike information criterion and Schwardz information criterion parameters; and finally, the presence of significant bidirectional and unidirectional spillover effects in the majority of instances. These findings are valuable for investors, regulators, and policymakers in enhancing returns and mitigating risk through portfolio diversification and informed decision making.

Active fluctuations in the harmonic chain: phonons, entropons and velocity correlations

Non-equilibrium random fluctuations of a non-thermal nature are a defining feature of active matter. In this work, we study the collective excitations of active systems at high density, focusing on a one-dimensional chain of elastically coupled inertial particles, where activity is modeled by an Ornstein–Uhlenbeck process. The excitation spectrum reveals two types of fluctuations: thermally excited phonons, analogous to those in passive crystals, and entropons, which are associated with entropy production due to active forces. These fluctuations exhibit distinct properties: only entropons generate spatial velocity correlations and violate the standard fluctuation-response relation. We derive exact expressions for equal-time velocity and displacement correlations, as well as for the structure factor, identifying the contributions from both phonons and entropons. Finally, we explore the dynamical properties of these excitations through steady-state two-time correlations, such as the intermediate scattering function and mean-square displacement. Both phonon and entropon fluctuations are characterized by a long-wavelength overdamped regime and a short-wavelength underdamped regime. In the large persistence case, entropons decay more slowly than phonons, and activity generally suppresses the oscillations typical of the underdamped regime.

The OU$^2$ process: Characterising dissipative confinement in noisy traps

Abstract The Ornstein-Uhlenbeck (OU) process describes the dynamics of Brownian particles in a confining harmonic potential, thereby constituting the paradigmatic model of overdamped, mean-reverting Langevin dynamics. Despite its widespread applicability, this model falls short when describing physical systems where the confining potential is itself subjected to stochastic fluctuations. However, such stochastic fluctuations generically emerge in numerous situations, including in the context of colloidal manipulation by optical tweezers, leading to inherently out-of-equilibrium trapped dynamics. To explore the consequences of stochasticity at this level, we introduce a natural extension of the OU process, in which the stiffness of the harmonic potential is itself subjected to OU-like fluctuations. We call this model the OU$^2$ process. We examine its statistical, dynamic, and thermodynamic properties through a combination of analytical and numerical methods. Importantly, we show that the probability density for the particle position presents power-law tails, in contrast to the Gaussian decay of the standard OU process. In turn, this causes the trapping behavior, extreme value statistics, first passage statistics, and entropy production of the OU$^2$ process to differ qualitatively from their standard OU counterpart. Due to the wide applicability of the standard OU process and of the proposed OU$^2$ generalisation, our study sheds light on the peculiar properties of stochastic dynamics in random potentials and lays the foundation for the refined analysis of the dynamics and thermodynamics of numerous experimental systems.

Detecting rough volatility: a filtering approach

In this paper, we focus on filtering and parameter estimation in stochastic volatility models when observations arise from high-frequency data. We are particularly interested in rough volatility models where spot volatility is driven by fractional Brownian motion with Hurst index H < 1 2 . Since volatility is not directly observable, we rely on particle filtering techniques for statistical inference regarding the current level of volatility and the parameters governing its dynamics. In order to obtain numerically efficient and recursive algorithms, we use the fact that a fractional Brownian motion can be represented through a superposition of Markovian semimartingales (Ornstein-Uhlenbeck processes). We analyze the performance of our approach on simulated data and we compare it to similar studies in the literature. The paper concludes with an empirical case study, where we apply our methodology to high-frequency data of a liquid stock.

A class of processes defined in the white noise space through generalized fractional operators

The generalization of fractional Brownian motion in the white and grey noise spaces has been recently carried over, following the Mandelbrot–Van Ness representation, through Riemann–Liouville type fractional operators. Our aim is to extend this construction by means of more general fractional derivatives and integrals, which we define as Fourier-multiplier operators and then specialize by means of Bernstein functions. More precisely, we introduce a general class of kernel-driven processes which encompasses, as special cases, a number of models in the literature, including fractional Brownian motion, tempered fractional Brownian motion, Ornstein–Uhlenbeck process. The greater generality of our model, with respect to the previous ones, allows a higher flexibility and a wider applicability. We derive here some properties of this class of processes (such as continuity, occupation density, variance asymptotics and persistence) according to the conditions satisfied by the Fourier symbol of the operator or the Bernstein function chosen. On the other hand, these processes are proved to display short- or long-range dependence, if obtained by means of a derivative or an integral type operator, respectively, regardless of the kernel used in their definition. Finally, this kind of construction allows us to define the corresponding noise and to solve a Langevin-type integral equation.

Local powers of least‐squares‐based test for panel fractional Ornstein–Uhlenbeck process

In recent years, significant advancements have been made in the field of identifying financial asset price bubbles, particularly through the development of time‐series unit‐root tests featuring fractionally integrated errors and panel unit‐root tests. This study introduces an innovative approach for assessing the sign of the persistence parameter () within a panel fractional Ornstein‐Uhlenbeck process, based on the least squares estimator of . This method incorporates three distinct test statistics based on the Hurst parameter (), which can take values in the range of , be equal to , or fall within the interval of . The null hypothesis corresponds to . Based on a panel of continuous records of observations, the null asymptotic distributions are obtained when the time span () is fixed and the number of cross sections () goes to infinity. The power function of the tests is obtained under the local alternative where is close to zero in the order of . This alternative covers the departure from the unit root hypothesis from the explosive side, enabling the calculation of lower power in bubble tests. The hypothesis testing problem and the local power function are also considered when a panel of discrete‐sampled observations is available under a sequential limit, that is, the sampling interval shrinks to zero followed by the goes to infinity.

Research on LSTM-Based Maneuvering Motion Prediction for USVs

Maneuvering motion prediction is central to the control and operation of ships, and the application of machine learning algorithms in this field is increasingly prevalent. However, challenges such as extensive training time, complex parameter tuning processes, and heavy reliance on mathematical models pose substantial obstacles to their application. To address these challenges, this paper proposes an LSTM-based modeling algorithm. First, a maneuvering motion model based on a real USV model was constructed, and typical operating conditions were simulated to obtain data. The Ornstein–Uhlenbeck process and the Hidden Markov Model were applied to the simulation data to generate noise and random data loss, respectively, thereby constructing a sample set that reflects real experiment characteristics. The sample data were then pre-processed for training, employing the MaxAbsScaler strategy for data normalization, Kalman filtering and RRF for data smoothing and noise reduction, and Lagrange interpolation for data resampling to enhance the robustness of the training data. Subsequently, based on the USV maneuvering motion model, an LSTM-based black-box motion prediction model was established. An in-depth comparative analysis and discussion of the model’s network structure and parameters were conducted, followed by the training of the ship maneuvering motion model using the optimized LSTM model. Generalization tests were then performed on a generalization set under Zigzag and turning conditions to validate the accuracy and generalization performance of the prediction model.

Robustness of Hilbert space-valued stochastic volatility models

In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

Characterization and analysis of generalized grey incomplete gamma noise

The grey incomplete gamma distributions was established by one of the authors in a previous publication. In this article we use the Kondratiev characterization theorem to identify those via a suitable Laplace transform with holomorphic functions with suitable properties. We establish theorems for the integration and convergence of sequences of these distributions. As direct applications of these analytic tools we give the examples of Donsker's delta function, identify the time-derivative of the process as a suitable distribution and define the Gamma grey Ornstein–Uhlenbeck process.