Ornstein-Uhlenbeck Stochastic Process Research Articles

Stochastic Response of an Airfoil and Its Effects on Lco’s Behavior Under Stall Flutter Regime

In this work, we investigate the effect of noise on a classical two degree-of-freedom pitch-plunge aeroelastic system. The inlet velocity of the flow is modelled as a stochastically varying parameter by the Ornstein-Uhlenbeck (OU) stochastic process. The system is a 2D airfoil and the elastic problem is simulated using linear springs. We study the manifestation of Limit Cycle Oscillations that corresponds to the varying fluid velocity under the dynamic stall regime. I aim to delve into the unexplored facets of the classical pitch-plunge aeroelastic system, seeking a comprehensive understanding of how parametric noise influences the occurrence of LCO and expands the boundaries of its known behavior.

Non-equilibrium Onsager–Machlup theory

This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or ‘piecewise’) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein–Uhlenbeck stochastic process (which is globally stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager–Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager–Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the non-equilibrium self-consistent generalised Langevin equation (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.

A Stochastic Model of Anomalously Fast Transport of Heat Energy in Crystalline Bodies

In this work, a new method for constructing the infinite-dimensional Ornstein–Uhlenbeck stochastic process is introduced. The constructed process is used to perturb the harmonic system in order to model anomalously fast heat transport in one-dimensional nanomaterials. The introduced method made it possible to obtain a transition probability function that allows for a different approach to the analysis of equations with such a disturbance. This creates the opportunity to relax assumptions about temporal correlations for such a process, which may lead to a qualitatively different model of energy transport through vibrations of the crystal lattice and, as a result, to obtain the superdiffusion equation on a macroscopic scale with an order of the fractional Laplacian different from the value of 3/4 obtained so far in stochastic models. Simulations confirming these predictions are presented and discussed.

The Fokker–Planck equation for the time-changed fractional Ornstein–Uhlenbeck stochastic process

In this paper, we study some properties of the generalized Fokker–Planck equation induced by the time-changed fractional Ornstein–Uhlenbeck process. First of all, we exploit some sufficient conditions to show that a mild solution of such equation is actually a classical solution. Then, we discuss an isolation result for mild solutions. Finally, we prove the weak maximum principle for strong solutions of the aforementioned equation and then a uniqueness result.

Ornstein-Uhlenbeck Process via Conflated Drive of Brownian Motion and Lévy Process and its Application

Non-linear time series and linear models were not designed to detect probabilistic process that are depict by velocity and drift associated to returns the way Ornstein-Uhlenbeck stochastic process describes diffusion and velocity associated to series or waves influenced by Brownian motion or Levy process. In this research, Brownian motion and Levy process were conflated as driving force for Ornstein-Uhlenbeck process with its solution applied to Naira-Dollar exchange rates from 2009-2019.The drift and diffusion estimates for the Ornstein-Uhlenbeck process driven by Brownian motion and Levy process are realization of AR (1) with 2.991 and 0.1672 respectively. The AR(1) realization for the Ornstein-Uhlenbeck process was stationary with estimate that lies outside the unit circle. The AIC, BIC, RMSE, and MSE for the Ornstein-Uhlenbeck process were estimated to be 483.7572, 483.4782, 0.00101, and 8.395 respectively, compare to estimates of the same indexes for AR (1) of 767.5, 634.09, 0.3819, and 23.48. The criterion via the residuals from the Ornstein-Uhlenbeck process was smaller, which connotes that the errors approximated in using drift, Brownian motion and to estimate is relatively small via the Ornstein-Uhlenbeck process. Keywords: Brownian motion; Drift; Diffusion; Levy process; Ornstein-Uhlenbeck Process DOI: 10.7176/MTM/11-3-02 Publication date: May 31 st 2021

Estimating Wind Farm Transformers Rating through Lifetime Characterization Based on Stochastic Modeling of Wind Power

This paper deals with the problem of the optimal rating of mineral-oil-immersed transformers in large wind farms. The optimal rating is derived based on the probabilistic analyses of wind power generation through the Ornstein–Uhlenbeck stochastic process and on thermal model of the transformer through the integration of stochastic differential equations. These analyses allow the stochastic characterization of lifetime reduction of the transformer and then its optimal rating through a simple closed form. The numerical application highlights the effectiveness and easy applicability of the proposed methodology. The proposed methodology allows deriving the rating of transformers which better fits the specific peculiarities of wind power generation. Compared to the conventional approaches, the proposed method can better adapt the transformer size to the intermittence and variability of the power generated by wind farms, thus overcoming the often-recognized reduced lifetime.

The role of grain-environment heterogeneity in normal grain growth: A stochastic approach

The size distribution of grains is a fundamental characteristic of polycrystalline solids. In the absence of deformation, the grain-size distribution is controlled by normal grain growth. The canonical model of normal grain growth, developed by Hillert, predicts a grain-size distribution that bears a systematic discrepancy with observed distributions. To address this, we propose a change to the Hillert model that accounts for the influence of heterogeneity in the local environment of grains. In our model, each grain evolves in response to its own local environment of neighbouring grains, rather than to the global population of grains. The local environment of each grain evolves according to an Ornstein-Uhlenbeck stochastic process. Our results are consistent with accepted grain-growth kinetics. Crucially, our model indicates that the size of relatively large grains evolves as a random walk due to the inherent variability in their local environments. This leads to a broader grain-size distribution than the Hillert model and indicates that heterogeneity has a critical influence on the evolution of the microstructure.

A Stochastic Model for Adsorption Kinetics

A novel stochastic model is proposed to characterize the adsorption kinetics of pollutants including dyes (direct red 80 and direct blue 1), fluoride ions, and cadmium ions removed by calcium pectinate (Pec-Ca), aluminum xanthanate (Xant-Al), and reed leaves, respectively. The model is based on a transformation over time following the Ornstein–Uhlenbeck stochastic process, which explicitly includes the uncertainty involved in the adsorption process. The model includes stochastic versions of the pseudo-first-order (PFO), pseudo-second-order (PSO), and pseudo-[Formula: see text]-order (PNO) models. It also allows the estimation of the adsorption parameters, including the maximum removal capacity ([Formula: see text]), the adsorption rate constant ([Formula: see text]), the reaction pseudoorder ([Formula: see text]), and the variability [Formula: see text]. The model fitted produced [Formula: see text] values similar to those of the nonstochastic versions of the PFO, PSO, and PNO models; however, the obtained values for each parameter indicate that the stochastic model better reproduces the experimental data. The [Formula: see text] values of the Pec-Ca-dye, Xant-Al-fluoride, and reed leaf-Cd+2 systems ranged from 2.0 to 9.7, 0.41 to 1.9, and 0.04 and 0.29 mg/g, respectively, whereas the values of [Formula: see text] ranged from 0.051 to 0.286, 0.743 to 75.73, and 0.756 to 8.861 (mg/g)1- n/min, respectively. These results suggest a variability in the parameters [Formula: see text] and [Formula: see text] inherent to the natures of the adsorbate and adsorbent. The obtained [Formula: see text] values ranged from 1.13 to 2.02 for the Pec-Ca-dye system, 1.0–3.5 for the Xant-Al-fluoride system, and 1.8–3.8 for the reed leaf-Cd+2 system. These ranges indicate the flexibility of the stochastic model to obtain fractional [Formula: see text] values, resulting in high [Formula: see text] values. The variability in each system was evaluated based on [Formula: see text]. The developed model is the first to describe pollutant removal kinetics based on a stochastic differential equation.

Active matter: Quantifying the departure from equilibrium.

Active matter systems are driven out of equilibrium at the level of individual constituents. One widely studied class are systems of athermal particles that move under the combined influence of interparticle interactions and self-propulsions, with the latter evolving according to the Ornstein-Uhlenbeck stochastic process. Intuitively, these so-called active Ornstein-Uhlenbeck particle (AOUP) systems are farther from equilibrium for longer self-propulsion persistence times. Quantitatively, this is confirmed by the increasing equal-time velocity correlations (which are trivial in equilibrium) and by the increasing violation of the Einstein relation between the self-diffusion and mobility coefficients. In contrast, the entropy production rate, calculated from the ratio of the probabilities of the position space trajectory and its time-reversed counterpart, has a nonmonotonic dependence on the persistence time. Thus, it does not properly quantify the departure of AOUP systems from equilibrium.

Business cycle modeling between financial crises and black swans: Ornstein-Uhlenbeck stochastic process vs Kaldor deterministic chaotic model.

Business cycles are oscillations in the economy because of recessions and expansions. In this paper we investigate the oscillation of the gross domestic product as a result of its relations with the other main macroeconomic variables such as capital, consumption, and investment. There is a long-standing debate about chaos and non-linear dynamics in economy and even the usefulness of those concepts has been questioned. Stochastic modeling has proven to be able to simulate reality fairly well. However, a stochastic behavior implies that reality is about exogenous randomness, while a chaotic behavior means that reality is deterministic and non-linearities are endogenous. Here we compare an Ornstein-Uhlenbeck stochastic process with a Kaldor-Kalecki deterministic chaotic model to understand which one fits better real data. We show that our chaotic model is able to represent reality as well as the stochastic model taken into consideration. Furthermore, our model may reproduce an extreme event (black swans).

Prediction of tensile response of UHPC with aligned and ZnPh treated steel fibers based on a spatial stochastic process

This paper proposed a new method for predicting tensile response of UHPC based on a spatial stochastic strength distribution model. The steel fibers in UHPC were aligned by a developed device and chemically treated by ZnPh. A direct tension test was conducted to obtain the tensile response of UHPC with treated fibers. Image analysis was used to obtain the frequency of overall fiber orientation distribution. The spatial stochastic distribution model was proposed for cross sectional strength of UHPC based on the Ornstein-Uhlenbeck stochastic process. The model introduced some assumptions such as the correlation coefficient between the cracking strength and fiber bridging strength. Parametric analysis was conducted to obtain the crucial stochastic parameters in the model. Combining with the single fiber pullout model, tensile behavior of a single crack, and minimum crack spacing, the model predicted the strain hardening and multiple cracking behaviors with precise accuracy, as compared to experimental results.

Business Cycle Modeling Between Financial Crises and Black Swans: Ornstein-Uhlenbeck Stochastic Process versus Kaldor Deterministic Chaotic Model

Business cycles are oscillations in the economy because of recessions and expansions. In this paper we investigate the oscillation of the gross domestic product as a result of its relations with the other main macroeconomic variables such as capital, consumption, and investment.There is a long-standing debate about chaos and non-linear dynamics in economy and even the usefulness of those concepts has been questioned. Stochastic modeling has proven to be able to simulate reality fairly well. However, a stochastic behavior implies that reality is about exogenous randomness, while a chaotic behavior means that reality is deterministic and non-linearities are endogenous. Here we compare an Ornstein–Uhlenbeck stochastic process with a Kaldor–Kalecki deterministic chaotic model to understand which one fits better real data. We show that our chaotic model is able to represent reality as well as the stochastic model taken into consideration. Furthermore, our model may reproduce an extreme event (black swans).

Closed-form analytical solutions for options on agricultural futures with seasonality and stochastic convenience yield

Abstract Stochastic commodity price models play a significant role in the pricing and hedging of commodity derivatives and real assets. This paper proposes a commodity pricing model that extends the Ewald and Ouyang two-factor model by adding a time-varying function into the volatility term of convenience yield process to describe seasonality. The stochastic factors in the model are the spot price of the commodity is assumed to follow geometrical Brownian motion process with a stochastic drift, the instantaneous convenience yield follows a mean-reverting Ornstein-Uhlenbeck stochastic process with time-varying seasonal volatility. The model we propose is analytically tractable and allows us to derive closed-form pricing formulas for futures and options by using the Mellin transform method. In addition, we provide the experiments results to illustrate the important properties of commodity futures options with numerical table and graphs.

Application of Iterated Filtering for Parametric Estimation of Instantaneous Variance in the Case of Non-Gaussian Ornstein-Uhlenbeck Stochastic Volatility Processes

The article presents a method for parametric estimation of instantaneous variance in the case of non-Gaussian Ornstein-Uhlenbeck stochastic volatility process by means of the iterated filtering and realized variance estimator. The method is applied to realized variance of S&P500 index data. Empirical application is accompanied with simulation study to examine performance of the estimation technique.

A Surfeit of Studies: What Have We Learned from All the Box Turtle (Terrapene carolina and T. ornata) Home Range Studies?

Home range (HR) studies are a particularly common approach to investigations of animal habitat use, resource availability, and response to management manipulation such as relocations. Terrapene carolina (Eastern box turtle) and its sister taxon T. ornata (Ornate box turtle) are especially popular subjects of HR studies because they are relatively easily tracked. Terrapene HR studies have revealed a wide variation in HR sizes within and between populations, due to factors such as differences in ecoregion and analytical approach (e.g., minimum convex polygons, kernel analysis, bivariate normal, multivariate Ornstein–Uhlenbeck stochastic process, harmonic means). We performed a meta-analysis of the available literature, including unpublished work to avoid bias due to under-publication, to explore the causes for variation in HR size. We found 19 studies reporting T. carolina HR sizes and seven studies reporting T. ornata HR sizes; the resulting meta-analysis revealed patterns that are not visible in the individual studies. We found important differences between the species: female T. ornata had smaller HRs than males, whereas the opposite is true for T. carolina, and T. ornata HRs were influenced by ecoregion, while T. carolina HRs were not similarly influenced. Not surprisingly, we found that choice of analysis technique affected HR estimate; analyses using ellipses resulted in larger HR estimates than all the other techniques, while kernels were smaller than minimum convex polygons. Although not indicated by individual studies, our meta-analysis showed that the HRs of relocated T. carolina females were significantly larger than those of non-relocated females. Although the number of individual turtles in studies varied from three to 25, the sample size did not significantly affect HR size.

A closed-form pricing formula for vulnerable European options under stochastic yield spreads and interest rates

Abstract This paper develops a three-factor valuation model of vulnerable European options incorporating stochastic yield spreads and interest rates, which extends a constant yield spread and deterministic interest rate proposed in Hull and White [12]. The dynamics of the short-term interest rate are represented implicitly by a stochastic bond price process. Therefore, the stochastic factors in the model are the spot price of the underlying asset that follows a geometrical Brownian motion process, the yield spreads and the default-free unit discount bond that are modeled by a mean-reverting Ornstein-Uhlenbeck stochastic process. Furthermore, we exploit Mellin transform techniques to derive a closed-form solution for vulnerable European options under the Black-Scholes model, which is simply computed using the standard normal cumulative distribution function so that the pricing and hedging of vulnerable European options can be computed very accurately and rapidly. Numerical experiments demonstrate how credit risk and interest rate risk affect the prices of European options.

Stochastic Stability of Coupled Viscoelastic Systems Excited by Real Noise

The moment stochastic stability and almost-sure stochastic stability of two-degree-of-freedom coupled viscoelastic systems, under the parametric excitation of a real noise, are investigated through the moment Lyapunov exponents and the largest Lyapunov exponent, respectively. The real noise is also called the Ornstein-Uhlenbeck stochastic process. For small damping and weak random fluctuation, the moment Lyapunov exponents are determined approximately by using the method of stochastic averaging and a formulated eigenvalue problem. The largest Lyapunov exponent is calculated through its relation with moment Lyapunov exponents. The stability index, the stability boundaries, and the critical excitation are obtained analytically. The effects of various parameters on the stochastic stability of the system are then discussed in detail. Monte Carlo simulation is carried out to verify the approximate results of moment Lyapunov exponents. As an application example, the stochastic stability of a flexural-torsional viscoelastic beam is studied.

Brownian motion in non-equilibrium systems and the Ornstein-Uhlenbeck stochastic process

The Ornstein-Uhlenbeck stochastic process is an exact mathematical model providing accurate representations of many real dynamic processes in systems in a stationary state. When applied to the description of random motion of particles such as that of Brownian particles, it provides exact predictions coinciding with those of the Langevin equation but not restricted to systems in thermal equilibrium but only conditioned to be stationary. Here, we investigate experimentally single particle motion in a two-dimensional granular system in a stationary state, consisting of 1 mm stainless balls on a plane circular surface. The motion of the particles is produced by an alternating magnetic field applied perpendicular to the surface of the container. The mean square displacement of the particles is measured for a range of low concentrations and it is found that following an appropriate scaling of length and time, the short-time experimental curves conform a master curve covering the range of particle motion from ballistic to diffusive in accordance with the description of the Ornstein-Uhlenbeck model.

Quantifying the value of investing in distributed natural gas and renewable electricity systems as complements: Applications of discounted cash flow and real options analysis with stochastic inputs

One energy policy objective in the United States is to promote the adoption of technologies that provide consumers with stable, secure, and clean energy. Recent work provides anecdotal evidence of natural gas (NG) and renewable electricity (RE) synergies in the power sector, however few studies quantify the value of investing in NG and RE systems together as complements. This paper uses discounted cash flow analysis and real options analysis to value hybrid NG-RE systems in distributed applications, focusing on residential and commercial projects assumed to be located in the states of New York and Texas. Technology performance and operational risk profiles are modeled at the hourly level to capture variable RE output and NG prices are modeled stochastically as geometric Ornstein-Uhlenbeck (OU) stochastic processes to capture NG price uncertainty. The findings consistently suggest that NG-RE hybrid distributed systems are more favorable investments in the applications studied relative to their single-technology alternatives when incentives for renewables are available. In some cases, NG-only systems are the favorable investments. Understanding the value of investing in NG-RE hybrid systems provides insights into one avenue towards reducing greenhouse gas emissions, given the important role of NG and RE in the power sector.

The Gamma renewal process as an output of the diffusion leaky integrate-and-fire neuronal model.

Statistical properties of spike trains as well as other neurophysiological data suggest a number of mathematical models of neurons. These models range from entirely descriptive ones to those deduced from the properties of the real neurons. One of them, the diffusion leaky integrate-and-fire neuronal model, which is based on the Ornstein-Uhlenbeck (OU) stochastic process that is restricted by an absorbing barrier, can describe a wide range of neuronal activity in terms of its parameters. These parameters are readily associated with known physiological mechanisms. The other model is descriptive, Gamma renewal process, and its parameters only reflect the observed experimental data or assumed theoretical properties. Both of these commonly used models are related here. We show under which conditions the Gamma model is an output from the diffusion OU model. In some cases, we can see that the Gamma distribution is unrealistic to be achieved for the employed parameters of the OU process.