Fractional Brownian Motion Research Articles

Averaging Principle for Mckean–Vlasov SDEs Driven by Multiplicative Fractional Noise With Highly Oscillatory Drift Coefficient

ABSTRACTIn this paper, we study averaging principle for a class of McKean–Vlasov stochastic differential equations (SDEs) that contain multiplicative fractional noise with Hurst parameter 1/2 and highly oscillatory drift coefficient. Here the integral corresponding to fractional Brownian motion is the generalized Riemann–Stieltjes integral. Using Khasminskii's time discretization techniques, we prove that the solution of the original system strongly converges to the solution of averaging system as the times scale goes to zero in the supremum‐ and Hölder ‐ topologies, which are sharpen existing ones in the classical McKean–Vlasov SDEs framework.

Non-Gaussian behavior in fractional Laplace motion with drift

We study fractional Laplace motion (FLM) obtained from subordination of fractional Brownian motion (FBM) to a gamma process in the presence of an external drift that acts on the composite process or of an internal drift acting solely on the parental process. We derive the statistical properties of this FLM process and find that the external drift does not influence the mean-squared displacement, whereas the internal drift leads to normal diffusion, dominating at long times in the subdiffusive Hurst exponent regime. We also investigate the intricate properties of the probability density function (PDF), demonstrating that it possesses a central Gaussian region whose expansion in time is influenced by FBM's Hurst exponent. Outside of this region, the PDF follows a non-Gaussian pattern. The kurtosis of this FLM process converges toward the Gaussian limit at long times insensitive to the extreme non-Gaussian tails. Additionally, in the presence of the external drift, the PDF remains symmetric and centered at x=vt. In contrast, for the internal drift this symmetry is broken. The results of our computer simulations are fully consistent with the theoretical predictions. The FLM model is suitable for describing stochastic processes with a non-Gaussian PDF and long-ranged correlations of the motion. Published by the American Physical Society 2025

Increasing domain infill asymptotics for stochastic differential equations driven by fractional Brownian motion

Although statistical inference in stochastic differential equations (SDEs) driven by Wiener process has received significant attention in the literature, inference in those driven by fractional Brownian motion seems to have seen much less development in comparison, despite their importance in modelling long-range dependence. In this article, we consider both classical and Bayesian inference in such fractional Brownian motion-based SDEs, observed on the time domain [ 0 , T ] . In particular, we consider asymptotic inference for two parameters in this regard; a multiplicative parameter β associated with the drift function, and the so called ‘Hurst parameter’ H of the fractional Brownian motion, when T → ∞ . For unknown H, the likelihood does not lend itself amenable to the popular Girsanov form, rendering usual asymptotic development difficult. As such, we develop increasing domain infill asymptotic theory, by discretizing the SDE into n discrete time points in [ 0 , T ] , and letting T → ∞ , n → ∞ , such that either n / T 2 or n/T tends to infinity. In this setup, we establish consistency and asymptotic normality of the maximum likelihood estimators, as well as consistency and asymptotic normality of the Bayesian posterior distributions. However, classical or Bayesian asymptotic normality with respect to the Hurst parameter could not be established. We supplement our theoretical investigations with simulation studies in a non-asymptotic setup, prescribing suitable methodologies for classical and Bayesian analyses of SDEs driven by fractional Brownian motion. Applications to a real, close price data, along with comparison with standard SDE driven by Wiener process, are also considered. As expected, it turned out that our Bayesian fractional SDE outperformed the other model and methods, in both simulated and real data applications.

Evidence and origin of anomalous diffusion of Ag+ ion in amorphous silica: A molecular dynamics study with neural network interatomic potentials.

The release of Ag+ ions into the environment through silica layers is a promising strategy for the development of anti-microbial surface coating devices. The aim of the present study is to provide some insight into the elementary mechanisms of diffusion of Ag+ ions through silica with the objective of proposing control strategies. Thanks to the development of interaction potentials based on neural networks, the diffusion processes were studied via molecular dynamics simulations. Silver diffusion was found to be anomalous and sub-diffusive, the origin of which could be attributed to deceleration and temporal anti-correlations. This sub-diffusion has been attributed primarily to the disordered nature of the silica matrix. Furthermore, it is magnified by the presence of coordination defects within the silica matrix. These defects, in particular the under-coordinated oxygen atoms, act as traps for Ag+ by forming O-Ag bonds, thereby limiting the jump length and retaining the ion for long duration. By comparison with existing diffusion models, the diffusion mechanism in the absence of defects appears to be of the fractional Brownian motion type, substantially modified by the presence of defects. Two possible approaches have emerged to tune the release of Ag+ ions through the silica layer: the monitoring of the number of defects and the opening/closing of diffusion paths via, e.g., a modification of the silica density.

Total controllability of stochastic non-instantaneous impulsive Hilfer fractional switched dynamic systems with deviated arguments

. This article investigates the solvability and controllability of stochastic non-instantaneous impulsive Hilfer fractional switched differential equations with deviated arguments and fractional Brownian motion (fBm) in the finite-dimensional space. The first part focuses on analyzing the existence and uniqueness of solutions using the Banach fixed-point theorem. In the second part, controllability results are established for the considered system. This study introduces a new class of control functions designed to govern the system at the termination of time intervals and on each impulsive event, incorporating stochastic noise. This approach leads to comprehensive controllability outcomes, often termed as total controllability results. The theoretical results are primarily established using fixed-point theorem, fractional calculus, Laplace transform, stochastic analysis, and Mittag-Leffler function. A numerical example is provided to validate the obtained theoretical results.

On the Long Range Dependence of Time-Changed Generalized Mixed Fractional Brownian Motion

We introduce a generalized mixed fractional Brownian motion (gmfBm) as a linear combination of n independent fractional Brownian motions with possibly different Hurst indices and investigate conditions under which the time-changed gmfBm exhibit long-range dependence when the time-change is induced by a tempered stable subordinator or a Gamma process. AMS Subject Classification (2020): Primary 60G22

Barrier-crossing driven by fractional Gaussian noise in the context of reactive flux formalism: An exact result

Barrier-crossing driven by fractional Gaussian noise in the context of reactive flux formalism: An exact result

Development of a computational model for variable-order fractional Brownian motion and solving associated stochastic integral equations using barycentric rational interpolants

Development of a computational model for variable-order fractional Brownian motion and solving associated stochastic integral equations using barycentric rational interpolants

Dynamical large deviations of the fractional Ornstein-Uhlenbeck process

Abstract The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation x ˙ ( t ) + γ x = 2 D ξ ( t ) , where ξ ( t ) is the fractional Gaussian noise with the Hurst exponent 0 < H < 1 . For H ≠ 1 / 2 the fOU process is non-Markovian but Gaussian, and it has either vanishing (for H < 1 / 2 ), or divergent (for H > 1 / 2 ) spectral density at zero frequency. For H > 1 / 2 , the fOU is long-correlated. Here we study dynamical large deviations of the fOU process and focus on the area A n = ∫ − T T x n ( t ) d t , n = 1 , 2 , … over a long time window 2T. Employing the optimal fluctuation method, we determine the optimal path of the conditioned process, which dominates the large-An tail of the probability distribution of the area, P ( A n , T ) ∼ exp ⁡ [ − S ( A n , T ) ] . We uncover a nontrivial phase diagram of scaling behaviors of the optimal paths and of the action S ( A n ≡ 2 a n T , T ) ∼ T α ( H , n ) a n 2 / n on the (H, n) plane. The phase diagram includes three distinct regions: (i) H > 1 − 1 / n , where α ( H , n ) = 2 − 2 H , and the optimal paths are delocalized in time, (ii) n = 2 and H ⩽ 1 2 , where α ( H , n ) = 1 , and the optimal paths oscillate with an H-dependent frequency, and (iii) H ⩽ 1 − 1 / n and n > 2, where α ( H , n ) = 2 / n , and the optimal paths are strongly localized. We verify our theoretical predictions in large-deviation simulations of the fOU process. By combining the Wang-Landau Monte-Carlo algorithm with the circulant embedding method of generation of stationary Gaussian fields, we were able to measure probability densities as small as 10−170. We also generalize our findings to other stationary Gaussian processes with either divergent, or vanishing spectral density at zero frequency.

Least squares estimation for fractional Brownian bridge with linear drift

. In this article, we research the asymptotic properties of least squares estimation for fractional Brownian bridge with linear drift d X t = α ( θ − X t ) T − t d t + d B t H , 0 ≤ t < T , with X 0 = x 0, where α > 0 , θ ∈ R and γ≜α θ are unknown parameters, B H is fractional Brownian motion with Hurst index H ∈ ( 1 / 2 , 1 ) , as well as T > 0. Based on the continuous-time observation, we obtain the estimators α ̂ , γ ̂ and θ ̂ for α, γ, and θ. Depending on the value of α, we can have the strong consistency or not as t→T. Since these estimates do not formally depend on the value of θ, we also get the rate of these convergences when the consistency holds in both θ≠0 and θ = 0 case. This work extends the results of Es-Sebaiy and Nourdin (2013) studied in the case where θ = 0 is known.

Limit theorems for some even-power integral functionals driven by fractional Brownian motion and fractional Brownian bridge

In this paper, we introduce some limit theorems for the even-power integral functionals ( 2 kβ + 1 ) ∫ 0 1 t 2 kβ ( W t H ) 2 k d t , ( α + 1 ) ∫ 0 1 t ( 2 kH + 1 ) α ( W t H ) 2 k d t , as β → + ∞ and α ↓ − 1 , where W H = { W t H , 0 ≤ t ≤ 1 } is a fractional Brownian motion with Hurst index 0<H<1 and k ≥ 1 is an integer.

Invariant submanifolds for solutions to rough differential equations

In this paper, we provide necessary and sufficient conditions for invariance of finite dimensional submanifolds for rough differential equations (RDEs) with values in a Banach space. Furthermore, we apply our findings to the particular situation of random RDEs driven by [Formula: see text]-Wiener processes and random RDEs driven by [Formula: see text]-fractional Brownian motion.

Stochastic volatility modeling via mixed fractional Brownian motion

Abstract The purpose of this work is to use the mixed fractional Brownian motion with parameter H ∈ ( 0 , 1 2 ) {H\in(0,\frac{1}{2})} as a modeling tool in financial mathematic, specifically in stochastic volatility modeling. We employ high frequency data from Oxford-Man Institute of Quantitative Finance Realized Library to estimate the smoothness of log-volatility. We show through empirical experiments that the measured smoothness of log-volatility is similar to that of mixed fractional Brownian motion when 0.09 ≤ H ≤ 0.2 {0.09\leq H\leq 0.2} . We propose to model the log-volatility by a mixed Fractional Brownian motion with H &lt; 1 2 {H&lt;\frac{1}{2}} and we construct a stationary mixed fractional Ornstein–Uhlenbeck process d ⁢ X t = - λ ⁢ ( X t - μ ) ⁢ d ⁢ t + γ ⁢ d ⁢ M t H , t ∈ [ 0 , T ] {dX_{t}=-\lambda(X_{t}-\mu)dt+\gamma dM_{t}^{H},t\in[0,T]} as a stationary model of log-volatility X t {X_{t}} and we show that it tends in distribution to the mixed fractional Brownian motion when λ → 0 {\lambda\rightarrow 0} .

European option pricing under a generalized fractional Brownian motion Heston exponential Hull–White model with transaction costs by the Deep Galerkin Method

European option pricing under a generalized fractional Brownian motion Heston exponential Hull–White model with transaction costs by the Deep Galerkin Method

Riemann-Liouville fractional Brownian motion with random Hurst exponent.

We examine two stochastic processes with random parameters, which in their basic versions (i.e., when the parameters are fixed) are Gaussian and display long-range dependence and anomalous diffusion behavior, characterized by the Hurst exponent. Our motivation comes from biological experiments, which show that the basic models are inadequate for accurate description of the data, leading to modifications of these models in the literature through introduction of the random parameters. The first process, fractional Brownian motion with random Hurst exponent (referred to as FBMRE below) has been recently studied, while the second one, Riemann-Liouville fractional Brownian motion with random exponent (RL FBMRE) has not been explored. To advance the theory of such doubly stochastic anomalous diffusion models, we investigate the probabilistic properties of RL FBMRE and compare them to those of FBMRE. Our main focus is on the autocovariance function and the time-averaged mean squared displacement of the processes. Furthermore, we analyze the second moment of the increment processes for both models, as well as their ergodicity properties. As a specific case, we consider the mixture of two-point distributions of the Hurst exponent, emphasizing key differences in the characteristics of RL FBMRE and FBMRE, particularly in their asymptotic behavior. The theoretical findings presented here lay the groundwork for developing new methods to distinguish these processes and estimate their parameters from experimental data.

A study of anomalous stochastic processes via generalizing fractional calculus.

Due to the very importance of fractional calculus in studying anomalous stochastic processes, we systematically investigate the existing formulation of fractional calculus and generalize it to broader applied contexts. Specifically, based on the improved Riemann-Liouville fractional calculus operators and the modified Maruyama's notation for fractional Brownian motion, we develop the fractional Ito^'s calculus and derive a generalized Fokker-Planck equation corresponding to the Maruyama's process, along with which, the stochastic realizations of trajectories, both underdamped and overdamped, have been studied in terms of the stochastic dynamics equations newly formulated. This paves a way to study the path integrals and the stochastic thermodynamics of anomalous stochastic processes. We also explicitly derive several fundamental results in fractional calculus, including the relation between fractional and normal differentiation, the Laplace transform for fractional derivatives, the analytic solution of one type of generalized diffusion equations, and the fractional integration formulas. Our results advance the existing fractional calculus and provide practical references for studying anomalous diffusion, mechanics of memory materials in engineering, and stochastic analysis in fractional orders.

Parameter identification of the Black-Scholes model driven by multiplicative fractional Brownian motion

Parameter identification of the Black-Scholes model driven by multiplicative fractional Brownian motion

On the increments of some extensions of the fractional Brownian motion

On the increments of some extensions of the fractional Brownian motion

Core Mass Function in View of Fractal and Turbulent Filaments and Fibers

Abstract We propose that the core mass function (CMF) can be driven by filament fragmentation. To model a star-forming system of filaments and fibers, we develop a fractal and turbulent tree with a fractal dimension of 2 and a Larson's law exponent $\beta$ of 0.5. The fragmentation driven by convergent flows along the splines of the fractal tree yields a Kroupa-IMF-like CMF that can be divided into three power-law segments with exponents $\alpha$ = -0.5, -1.5, and -2, respectively. The turnover masses of the derived CMF are approximately four times those of the Kroupa IMF, corresponding to a star formation efficiency of 0.25. Adopting $\beta$=1/3, which leads to fractional Brownian motion along the filament, may explain a steeper CMF at the high-mass end, with $\alpha$=-3.33 close to that of the Salpeter IMF. We suggest that the fibers of the tree are basic building blocks of star formation, sharing similar properties across different clouds, setting a common density threshold for star formation and leading to a universal CMF.

Regularity and strong convergence of numerical approximations for stochastic wave equations with multiplicative fractional Brownian motions

Regularity and strong convergence of numerical approximations for stochastic wave equations with multiplicative fractional Brownian motions