Fractional Ornstein-Uhlenbeck Research Articles

Gas fees on the Ethereum blockchain: from foundations to derivative valuations

The “gas fee” paid for inclusion in the blockchain is analyzed in two parts. First, we consider how “effort” in terms of resources required to process and store a transaction turns into a “gas limit,” which, through a fee comprised of the “base” and “priority fee” in the current version of Ethereum, is converted into the cost paid by the user. We adhere closely to the Ethereum protocol to simplify the analysis and to constrain the design choices when considering “multidimensional gas.” Second, we assume that the “gas” price is given deus ex machina by a fractional Ornstein–Uhlenbeck process and evaluate various derivatives. These contracts can, for example, mitigate gas cost volatility. The ability to price and trade “forwards” in addition to the existing “spot” inclusion into the blockchain could enable users to hedge against future cost fluctuations. Overall, this article offers a comprehensive analysis of gas fee dynamics on the Ethereum blockchain, integrating supply-side constraints with demand-side modelling to enhance the predictability and stability of transaction costs.

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 $H<1/2$.


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.

Deep learning the Hurst parameter of linear fractional processes and assessing its reliability

AbstractThis research explores the reliability of deep learning, specifically Long Short‐Term Memory (LSTM) networks, for estimating the Hurst parameter in fractional stochastic processes. The study focuses on three types of processes: fractional Brownian motion (fBm), fractional Ornstein–Uhlenbeck (fOU) process, and linear fractional stable motions (lfsm). The work involves a fast generation of extensive datasets for fBm and fOU to train the LSTM network on a large volume of data in a feasible time. The study analyses the accuracy of the LSTM network's Hurst parameter estimation regarding various performance measures like root mean squared error (RMSE), mean absolute error (MAE), mean relative error (MRE), and quantiles of the absolute and relative errors. It finds that LSTM outperforms the traditional statistical methods in the case of fBm and fOU processes; however, it has limited accuracy on lfsm processes. The research also delves into the implications of training length and valuation sequence length on the LSTM's performance. The methodology is applied to estimating the Hurst parameter in li‐ion battery degradation data and obtaining confidence bounds for the estimation. The study concludes that while deep learning methods show promise in parameter estimation of fractional processes, their effectiveness is contingent on the process type and the quality of training data.

An active fractional Ornstein–Uhlenbeck particle: diffusion and dissipation

In this work, we consider a particle subjected to active fluctuations due to the presence of an active bath. The active fluctuation driving the particle is modeled as an Ornstein–Uhlenbeck process with memory, due to the recent experimental result in that a bacterial bath under a chemotactic mechanism exhibits a large distribution of persistence time, which may result in long-range persistence of a probe particle that cannot be captured by the conventional Ornstein–Uhlenbeck model for active noise. As a paradigmatic model, we consider introducing a fractional Ornstein–Uhlenbeck process with a power-law kernel, which includes a fractional order variable 0<α⩽1 , to model an active noise acting on a particle. We present the relevant properties of the active noise and its implications for the diffusion of free and harmonically confined single particles. This is also supported by the derived active Smoluchowski description. More importantly, we find that the effect of α can either reduce or increase the effective temperature at long time, depending on the competition between the harmonic timescale and persistence time. Lastly, the effects of presented active noise on the spectral density of the energy dissipation rate and the dissipation energy from both the thermal and active bath are explored.

PSD and Cross-PSD of Responses of Seven Classes of Fractional Vibrations Driven by fGn, fBm, Fractional OU Process, and von Kármán Process

This paper gives its contributions in four stages. First, we propose the analytical expressions of power spectrum density (PSD) responses and cross-PSD responses to seven classes of fractional vibrators driven by fractional Gaussian noise (fGn). Second, we put forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by fractional Brownian motion (fBm). Third, we present the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators driven by the fractional Ornstein–Uhlenbeck (OU) process. Fourth, we bring forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by the von Kármán process. We show that the statistical dependences of the responses to seven classes of fractional vibrators follow those of the excitation of fGn, fBm, the OU process, or the von Kármán process. We also demonstrate the obvious effects of fractional orders on the responses to seven classes of fractional vibrations. In addition, we newly introduce class VII fractional vibrators, their frequency transfer function, and their impulse response in this research.

Fractional stochastic volatility model

This article introduces a discrete‐time fractional stochastic volatility model (FSV) based on fractional Gaussian noise. The new model includes the standard stochastic volatility model as a special case and has the same limit as the fractional integrated stochastic volatility (FISV) model, which is the continuous‐time fractional Ornstein–Uhlenbeck process. A simulated maximum likelihood method, which maximizes the time‐domain log‐likelihood function calculated by the importance sampling technique, and a frequency‐domain quasi maximum likelihood method (or quasi Whittle) are employed to estimate the model parameters. Simulation studies suggest that, while both estimation methods can accurately estimate the model, the simulated maximum likelihood method outperforms the quasi Whittle method. As an illustration, we fit the FSV and FISV models with the proposed estimation techniques to the S&amp;P 500 composite index over a sample period spanning 45 years.

Global attracting set of stochastic differential equations with unbounded delay driven by fractional Ornstein–Uhlenbeck process

Abstract 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 ) {{Y^{H,\xi}}(t)} with H ∈ ( 1 2 , 1 ) {H\in(\frac{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.

Variance and volatility swaps and options under the exponential fractional Ornstein–Uhlenbeck model

Considering the fair strike values of variance and volatility swaps, we use a stochastic volatility model in which the log volatility is given by a fractional Ornstein–Uhlenbeck process with two versions; a stationary version and a version with a deterministic initial value. Under these versions, the fair strike formulas are obtained in exact form for variance swaps and approximated fair strike formulas are derived for volatility swaps based on the fact that an aggregation of log-normal variables is well-approximated by shifted log-normal or log-normal distribution. In addition, we obtain two approximate pricing formulas for European options on the realized variance and volatility. The accuracy and robustness of the approximated fair strike formulas are examined via Monte-Carlo computations. We conduct calibration experiments to show that the Hurst exponent and the mean-reversion property of the fractional Ornstein–Uhlenbeck process are able to produce various shapes resembling the market term-structures of variance swaps when they are put together.

Ergodic criterion of a random diffusivity model.

The random diffusivity, initially proposed to explain Brownian yet non-Gaussian diffusion, has garnered significant attention due to its capacity not only for elucidating the internal physical mechanism of non-Gaussian diffusion, but also for establishing an analytical framework to characterize particle motion in complex environments. In this paper, based on the correlation function C(t_{1},t_{2})=〈D(t_{1})D(t_{2})〉 of random diffusivity D(t), we quantitatively propose a general criterion of determining the ergodic property of the Langevin equationwith the arbitrary random diffusivity D(t). Due to the critical role of correlation function C(t_{1},t_{2}), we derive the criterion for the two cases with stationary diffusivity or nonstationary diffusivity, respectively. By utilizing the quantitative criterion, we can directly judge the ergodic properties of the random diffusivity model based on the correlation function C(t_{1},t_{2}) of random diffusivity D(t). Several typical diffusivities, including the common square of the Brownian motion and of the (fractional) Ornstein-Uhlenbeck process, are found to contribute to different ergodic properties, which validates our proposed criterion built on the correlation function C(t_{1},t_{2}).

The El Niño Southern Oscillation Recharge Oscillator with the Stochastic Forcing of Long-Term Memory

The influence of the fast-varying variables that have a long-term memory on the El Niño Southern Oscillation (ENSO) is investigated by adding a fractional Ornstein–Uhlenbeck (FOU) process stochastic noise on the simple recharge oscillator (RO) model. The FOU process noise converges to zero very slowly with a negative power law. The corresponding non-zero ensemble mean during the integration period can exert a pronounced influence on the ensemble-mean dynamics of the RO model. The state-dependent noise, also called the multiplicative noise, can present its influence by reducing the relaxation coefficient and by introducing periodic external forcing. The decreasing relaxation coefficient can enhance the oscillation amplitude and shorten the oscillation period. The forced frequency is close to the natural frequency. The two mechanisms together can further amplify the amplitude and shorten the period, compared with the state-independent noise or additive noise, which only exhibits its influence by introducing non-periodic external forcing. These two mechanisms explicitly elucidate the influence of the stochastic forcing on the ensemble-mean dynamics of the RO model. It provides comprehensive knowledge to better understand the interaction between the fast-varying stochastic forcing and the slow-varying deterministic system and deserves further investigation.

The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift

The long time behavior of the fractional Ornstein-Uhlenbeck process with linear self-repelling drift

Asymptotic theory for explosive fractional Ornstein-Uhlenbeck processes

Asymptotic theory for explosive fractional Ornstein-Uhlenbeck processes

Quantile estimation in fractional Levy Ornstein-Uhlenbeck processes

First we study estimation of the drift parameter in the fractional Ornstein-Uhlenbeck process whose marginal distribution is Student t-distribution. We obtain Spearman’s correlation based estimator, quantile estimator and Brownian excursion based estimator of the drift parameter. Then we study method of moments estimator and quantile estimator in fractional inverse Gaussian and fractional gamma Ornstein-Uhlenbeck processes.

Moderate Deviations for Parameter Estimation in the Fractional Ornstein-Uhlenbeck Processes with Periodic Mean

Moderate Deviations for Parameter Estimation in the Fractional Ornstein-Uhlenbeck Processes with Periodic Mean

Asymptotically efficient estimation of Ergodic rough fractional Ornstein-Uhlenbeck process under continuous observations

Asymptotically efficient estimation of Ergodic rough fractional Ornstein-Uhlenbeck process under continuous observations

Statistical inference in discretely observed fractional Ornstein–Uhlenbeck processes

In this paper, we develop a two-stage method for estimating all the unknown parameters in the fractional Ornstein–Uhlenbeck model from discrete observations. The estimation procedure is built upon the marriage of the power variation and the minimum contrast estimation. In the first stage, the Hurst coefficient is estimated by the increment Bernoulli statistic and the volatility parameter is estimated by power variations. The asymptotic theory of the proposed estimators in the diffusion term are established under an in-fill asymptotic scheme. In the second stage, two drift parameters are estimated based on the minimum contrast estimation. Their asymptotic theory are analyzed via use of a double asymptotic scheme. The results from Monte Carlo studies illustrate that the proposed estimators have reasonable finite sample properties. An empirical illustration based on realized volatility indicates that the realized volatility is rough.

Parameter estimation of the fractional Ornstein-Uhlenbeck process based on quadratic variation.

Modern experiments routinely produce extensive data of the diffusive dynamics of tracer particles in a large range of systems. Often, the measured diffusion turns out to deviate from the laws of Brownian motion, i.e., it is anomalous. Considerable effort has been put in conceiving methods to extract the exact parameters underlying the diffusive dynamics. Mostly, this has been done for unconfined motion of the tracer particle. Here, we consider the case when the particle is confined by an external harmonic potential, e.g., in an optical trap. The anomalous particle dynamics is described by the fractional Ornstein-Uhlenbeck process, for which we establish new estimators for the parameters. Specifically, by calculating the empirical quadratic variation of a single trajectory, we are able to recover the subordination process governing the particle motion and use it as a basis for the parameter estimation. The statistical properties of the estimators are evaluated from simulations.

APPROXIMATE PRICING OF DERIVATIVES UNDER FRACTIONAL STOCHASTIC VOLATILITY MODEL

AbstractThis paper examines the issue of derivative pricing within the framework of a fractional stochastic volatility model. We present a deterministic partial differential equation system to derive an approximate expression for the derivative price. The proposed approach allows for the stochastic volatility to be expressed as a composition of deterministic functions of time and a fractional Ornstein–Uhlenbeck process. We apply this method to the European option pricing under the fractional Stein–Stein volatility model, demonstrating its feasibility and reliability through numerical simulations. Our numerical simulations also illustrate the impact of the parameters in the fractional stochastic volatility model on the option price.

Nonlinear biases in the roughness of a Fractional Stochastic Regularity Model

A Multifractional Process with Random Exponent (MPRE) is used to model the dynamics of log-prices in a financial market. Under this assumption, we show that the Hurst-Hölder exponent of the MPRE follows the fractional Ornstein–Uhlenbeck process which in the Fractional Stochastic Volatility Model of Comte and Renault (1998) describes the dynamics of the log-volatility. We provide evidence that several biases of the estimation procedures can generate artificial rough volatility in surrogated as well as real financial data.