Fractional Brownian Motion Research Articles

Machine-learning classification with additivity and diverse multifractal pathways in multiplicativity

Evidence of multifractal structures has spread to a wider set of physiological time series supporting the intricate interplay of biological and psychological functioning. These dynamics manifest as random multiplicative cascades, embodying nonlinear relationships characterized by recurring division, branching, and aggregation processes implicating noise across successive generations. This investigation focuses on how well the diversity of multifractal properties can be specific to the type of cascade relationship between generation (i.e., multiplicative, additive, or a mixture) as well as to the type of noise (i.e., including additive white Gaussian noise, fractional Gaussian noise, and various amalgamations) among 15 distinct types of binomial cascade processes. Cross-correlation analysis of multifractal spectral features confirms that these features capture nuanced aspects of cascading processes with minimal redundancy. Principal component analysis using 13 distinct multifractal spectral features shows that different cascade processes can manifest multifractal evidence of nonlinearity for distinct reasons. This transparency of multifractal spectral features to underlying cascade dynamics becomes less amenable to machine-learning strategies. Fully connected neural networks struggled to classify the 15 distinct types of cascade processes based on the respective multifractal spectral features (45.5% accuracy) yet demonstrated improved accuracy when addressing single categories of cross-generation relationships, that is, additive (91.6%), multiplicative (75.4%), or additomultiplicative (70.6%). While traditional principal component analysis reveals distinct loadings attributed to individual noise processes, multiplicative relationships between generations effectively make the constituent noise processes less discernible to neural networks. Neural networks may lack sufficient hierarchical depth required to effectively distinguish among nonadditive cascading processes, recommending either elaborating multifractal geometry or using alternate architectures for machine-learning classification of cascades with multiplicative relationships. Published by the American Physical Society 2024

Non-confluence for SDEs driven by fractional Brownian motion with Markovian switching

Non-confluence for SDEs driven by fractional Brownian motion with Markovian switching

Nanobubbles in Ultrapure Water Can Self-Propel.

Nanobubbles are sub- micron-sized gas entities that find applications in a wide range of scientific fields. Typically, they are thought to diffuse according to Brownian motion. We report the existence of self-propelled motion of oxygen bulk nanobubbles in ultrapure water at body temperature. Their motion, to a large extent, is self-affine; there are different scaling exponents along the x- and y-axes as well as for the lateral displacement. We use fractal analysis, and we calculate the structure function, the normalised velocity autocorrelation function, the skewness, and the kurtosis. All descriptors attest the existence of a quasi-Gaussian stochastic process, which is classified as fractional Brownian motion. More than 50 \% of the trajectories along the x-axis follow superdiffusion, while this amount drops to 30 \% for motion along the y-axis as a result of the asymmetry of the field of view.

Siegmund duality for physicists: a bridge between spatial and first-passage properties of continuous- and discrete-time stochastic processes

Abstract We consider a generic one-dimensional stochastic process x(t), or a random walk Xn , which describes the position of a particle evolving inside an interval [ a , b ] , with absorbing walls located at a and b. In continuous time, x(t) is driven by some equilibrium process θ ( t ) , while in discrete time, the jumps of Xn follow a stationary process that obeys a time-reversal property. An important observable to characterize its behavior is the exit probability E b ( x , t ) , which is the probability for the particle to be absorbed first at the wall b, before or at time t, given its initial position x. In this paper we show that the derivation of this quantity can be tackled by studying a dual process y(t) very similar to x(t) but with hard walls at a and b. More precisely, we show that the quantity E b ( x , t ) for the process x(t) is equal to the probability Φ ~ ( x , t | b ) of finding the dual process inside the interval [ a , x ] at time t, with y ( 0 ) = b . This is known as Siegmund duality in mathematics. Here we show that this duality applies to various processes that are of interest in physics, including models of active particles, diffusing diffusivity models, a large class of discrete- and continuous-time random walks, and even processes subjected to stochastic resetting. For all these cases, we provide an explicit construction of the dual process. We also give simple derivations of this identity both in the continuous and in the discrete time setting, as well as numerical tests for a large number of models of interest. Finally, we use simulations to show that the duality is also likely to hold for more complex processes, such as fractional Brownian motion.

Probability of entering an orthant by correlated fractional Brownian motion with drift: exact asymptotics

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Cameron–Martin type theorem for a class of non-Gaussian measures

. In this article, we study the quasi-translation-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron–Martin space and derive the explicit Radon-Nikodym density in terms of the Wiener integral with respect to the fractional Brownian motion. Moreover, we show an integration by parts formula for the derivative operator in the directions of the Cameron–Martin space. As a consequence, we derive the closability of both the derivative and the corresponding gradient operators.

Fractional Poisson and Gamma Models for Rainfall: Implications Climate Change and Marine Ecosystems

This research explores the benefits of using fractional Poisson and fractional Gamma models in rainfall modeling, highlighting their advantages in handling zero-inflated data, reducing overdispersion, and providing greater flexibility and accuracy. The second part of this study delves into the dynamic interplay between oceanic ecosystems and global climate change. It focuses on the role of phytoplankton in oxygen production and the impact of warming waters on this delicate balance. By employing mathematical models integrating differential equations and Brownian motion, the study offers a comprehensive framework for understanding how varying rates of oxygen production influence the sustainability of oceanic ecosystems. Finally, the research incorporates fractional Brownian motion into modeling plankton-oxygen dynamics, addressing the limitations of traditional Brownian motion. This approach captures the long-range dependencies and persistent effects critical for predicting the response of marine ecosystems to climate change. The findings underscore the need for nuanced mitigation strategies to address the imminent risks posed by global warming on marine life and atmospheric oxygen levels.

Boosting the HP filter for trending time series with long-range dependence

This article extends recent asymptotic theory developed for the Hodrick Prescott (HP) filter and boosted HP (bHP) filter to long-range dependent time series that have fractional Brownian motion (fBM) limit processes after suitable standardization. Under general conditions, it is shown that the asymptotic form of the HP filter is a smooth curve, analogous to the finding in Phillips and Jin for integrated time series and series with deterministic drifts. Boosting the filter using the iterative procedure suggested in Phillips and Shi leads under well-defined rate conditions to a consistent estimate of the fBM limit process or the fBM limit process with an accompanying deterministic drift when that is present. A stopping criterion is used to automate the boosting algorithm, giving a data-determined method for practical implementation. The theory is illustrated in simulations and two real data examples that highlight the differences between simple HP filtering and the use of boosting. The analysis is assisted by employing a uniformly and almost surely convergent trigonometric series representation of fBM.

On Some Stochastic Parabolic Systems Driven by New Fractional Brownian Motions

On Some Stochastic Parabolic Systems Driven by New Fractional Brownian Motions

Control and stochastic dynamic behavior of Fractional Gaussian noise-excited time-delayed inverted pendulum system

In this paper, we investigate the control and dynamic behavior of the inverted pendulum system with time delay under fractional Gaussian noise excitation. For H=1/2 and H∈(1/2,1), we analyze the stochastic dynamic characteristics of the system under Hopf bifurcation, utilizing time delay and noise intensity as bifurcation parameters, and validate the theoretical conclusions through numerical simulations. We also defined the engineering application range of angle and angular velocity under both asymptotically stable and periodic oscillation dynamic states. Furthermore, using the stochastic Itoˆ equation, we determined the values of time delay and noise intensity that satisfy the maximum engineering application range of angle and angular velocity, and verified their accuracy against the original equation. Additionally, we observed stochastic D-bifurcation and P-bifurcation arising from the combined effects of time delay and noise. Our results exhibit remarkable consistency between analytical and numerical findings, affirming the robustness of our approach and shedding light on the intricate dynamics of the system.

Incorporating memory effect into a fractional stochastic diffusion particle tracking model for suspended sediment using Malliavin-Calculus-based fractional Brownian Motion

This study aims to develop a Fractional Stochastic Diffusion Particle Tracking Model (FSDPTM) to illustrate suspended sediment’s long-term dependence (LTD) trajectories immersed in fully developed turbulent flow. Carried by eddies in the open channel, individual suspended sediment behavior exhibits dependence on its past increments. The Fractional Brownian Motion (FBM) is incorporated into the FSDPTM to account for the randomness of turbulent flow and to include the time-dependent increments as a memory effect. Additionally, to enhance the capacity for differentiating FBM, this study introduces Malliavin Calculus, the stochastic calculus of variation, to construct the square -differentiable FBM. For the numerical realizations of square differentiable FBM, the integration of fractional Gaussian white noise and Wiener–Ito Chaos expansion with the Hermite sequence is employed. The FSDPTM offers probabilistic LTD trajectories as solutions to the Ito-type Langevin equation. It also provides particle velocity, acceleration, and ensemble statistics, collectively demonstrating the application of FBM to sediment transport modeling. In conclusion, the key feature of the FSDPTM is its ability to translate LTD derived from FBM into the trajectory of suspended sediment particles in fully developed turbulent flow. Additionally, it expands the solutions from non-differentiable to square-differentiable stochastic processes.

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.

Spines and Inclines: Bioinspired Spines on an Insect-Scale Robot Facilitate Locomotion on Rough and Inclined Terrain.

To navigate complex terrains, insects use diverse tarsal structures (adhesive pads, claws, spines) to reliably attach to and locomote across substrates. This includes surfaces of variable roughness and inclination, which often require reliable transitions from ambulatory to scansorial locomotion. Using bioinspired physical models as a means for comparative research, our study specifically focused on the diversity of tarsal spines, which facilitate locomotion via frictional engagement and shear force generation. For spine designs, we took inspiration from ground beetles (Family Carabidae), which is a largely terrestrial group known for their quick locomotion. Evaluating four different species, we found that the hind legs host linear rows of rigid spines along the entire tarsus. By taking morphometric measurements of the spines, we highlighted parameters of interest (e.g., spine angle and aspect ratio) in order to test their relationship to shear forces sustained during terrain interactions. We systematically evaluated these parameters using spines cut from stainless steel shim attached to a small acrylic sled loaded with various weights. The sled was placed on 3D-printed models of rough terrain, randomly generated using fractal Brownian motion, while a motorized pulley system applied force to the spines. A force sensor measured the reaction force on the terrain, recording shear force before failure occurred. Initial shear tests highlighted the importance of spine angle, with bioinspired anisotropic designs producing higher shear forces. Using this data, we placed the best (50○ angle) and worst (90○ angle) performing spines on the legs of our insect-scale ambulatory robot physical model. We then tested the robot on various surfaces at 0, 10 and 20○ inclines, seeing similar success with the more bioinspired spines.

Response of MDOF nonlinear system with fractional derivative damping and driven by fractional Gaussian noise

The integration of fractional calculus into stochastic dynamics has introduced two innovative concepts, i.e., fractional derivative damping (FDD) and fractional Gaussian noise (fGn). To date, most studies on fractional stochastic dynamics have exclusively focused on either FDD or fGn, rather than concurrently exploring both. It is the first attempt for this paper to investigate a multi-degree-of-freedom (MDOF) nonlinear system with FDD and driven by fGn at the same time. Employing the generalized harmonic balance technique, FDD is equivalently decomposed into a quasi-linear restoring force and a quasi-linear damping force. Consequently, the original random Lagrangian system with FDD is transformed into the stochastically excited and dissipated Hamiltonian system. Utilizing the characteristic of the relatively flat power spectral density of fGn, it is approximated as wideband noise. This enables the application of the stochastic averaging method for quasi-integrable Hamiltonian system under wideband noise. Two detailed examples are carried out to provide the technical procedures and numerical verifications.

Besicovitch almost automorphic solutions in finite‐dimensional distributions to stochastic semilinear differential equations driven by both Brownian and fractional Brownian motions

Besicovitch almost automorphic solutions in finite‐dimensional distributions to stochastic semilinear differential equations driven by both Brownian and fractional Brownian motions

Understanding Short-Range Memory Effects in Deep Neural Networks.

Stochastic gradient descent (SGD) is of fundamental importance in deep learning. Despite its simplicity, elucidating its efficacy remains challenging. Conventionally, the success of SGD is ascribed to the stochastic gradient noise (SGN) incurred in the training process. Based on this consensus, SGD is frequently treated and analyzed as the Euler-Maruyama discretization of stochastic differential equations (SDEs) driven by either Brownian or Lévy stable motion. In this study, we argue that SGN is neither Gaussian nor Lévy stable. Instead, inspired by the short-range correlation emerging in the SGN series, we propose that SGD can be viewed as a discretization of an SDE driven by fractional Brownian motion (FBM). Accordingly, the different convergence behavior of SGD dynamics is well-grounded. Moreover, the first passage time of an SDE driven by FBM is approximately derived. The result suggests a lower escaping rate for a larger Hurst parameter, and thus, SGD stays longer in flat minima. This happens to coincide with the well-known phenomenon that SGD favors flat minima that generalize well. Extensive experiments are conducted to validate our conjecture, and it is demonstrated that short-range memory effects persist across various model architectures, datasets, and training strategies. Our study opens up a new perspective and may contribute to a better understanding of SGD.

Application of the Fractal Brownian Motion to the Athens Stock Exchange

The Athens Stock Exchange (ASE) is a dynamic financial market with complex interactions and inherent volatility. Traditional models often fall short in capturing the intricate dependencies and long memory effects observed in real-world financial data. In this study, we explore the application of fractional Brownian motion (fBm) to model stock price dynamics within the ASE, specifically utilizing the Athens General Composite (ATG) index. The ATG is considered a key barometer of the overall health of the Greek stock market. Investors and analysts monitor the index to gauge investor sentiment, economic trends, and potential investment opportunities in Greek companies. We find that the Hurst exponent falls outside the range typically associated with fractal Brownian motion. This, combined with the established non-normality of increments, disfavors both geometric Brownian motion and fractal Brownian motion models for the ATG index.

European Call Option under Stochastic Interest Rate in a Fractional Brownian Motion with Transaction Cost

This paper deals on the valuation of European call option price in a stochastic environment by employing three factors which are the stochastic model of the asset value, the stochastic interest rate and the transaction cost. We specify that our underlying asset and the stochastic interest rate, particularly Hull-White model, follows a fractional Brownian Motion governed by Hurst parameter H. We used the hedging and replicating technique to established the zero-coupon bond on the European option. Finally, we give a closed-form formula of the European call option price.

Existence of global and explosive mild solutions of fractional reaction–diffusion system of semilinear SPDEs with fractional noise

In this paper, we investigate the existence and finite-time blow-up for the solution of a reaction–diffusion system of semilinear stochastic partial differential equations (SPDEs) subjected to a two-dimensional fractional Brownian motion given by [Formula: see text] for [Formula: see text], along with [Formula: see text] where [Formula: see text] is the fractional power [Formula: see text] of the Laplacian, [Formula: see text] and [Formula: see text] and [Formula: see text] are real constants. We provide sufficient conditions for the existence of a global weak solution. Under the assumption that [Formula: see text] with Hurst index [Formula: see text] we obtain the blow-up times for an associated system of random partial differential equations in terms of an integral representation of exponential functions of Brownian motions. Moreover, we provide lower and upper bounds for the finite-time blow-up of the above system of SPDEs and obtain the upper bounds for the probability of non-explosive solution to our considered system.

Valuation of Vulnerable Barrier Options in a Mixed Fractional Brownian Motion Environment

Valuation of Vulnerable Barrier Options in a Mixed Fractional Brownian Motion Environment