Additive Fractional Noise Research Articles

Concentration inequalities for Stochastic Differential Equations with additive fractional noise

In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval [0, T ] of an additive SDE driven by a fractional Brownian motion with Hurst parameter H ∈ (0, 1) and for functionals of discrete-time observations of this process. Then, we apply this general result to specific functionals related to discrete and continuous-time occupation measures of the process.

LAN property for stochastic differential equations with additive fractional noise and continuous time observation

We consider a stochastic differential equation with additive fractional noise with Hurst parameter H>1∕2, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate τ as τ→∞, when the solution is observed continuously on the time interval [0,τ]. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyze the particular case of the fractional Ornstein–Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.

Memory effects for a stochastic fractional oscillator in a magnetic field

The problem of random motion of harmonically trapped charged particles in a constant external magnetic field is studied. A generalized three-dimensional Langevin equation with a power-law memory kernel is used to model the interaction of Brownian particles with the complex structure of viscoelastic media (e.g., dusty plasmas). The influence of a fluctuating environment is modeled by an additive fractional Gaussian noise. In the long-time limit the exact expressions of the first-order and second-order moments of the fluctuating position for the Brownian particle subjected to an external periodic force in the plane perpendicular to the magnetic field have been calculated. Also, the particle's angular momentum is found. It is shown that an interplay of external periodic forcing, memory, and colored noise can generate a variety of cooperation effects, such as memory-induced sign reversals of the angular momentum, multiresonance versus Larmor frequency, and memory-induced particle confinement in the absence of an external trapping field. Particularly in the case without external trapping, if the memory exponent is lower than a critical value, we find a resonancelike behavior of the anisotropy in the particle position distribution versus the driving frequency, implying that it can be efficiently excited by an oscillating electric field. Similarities and differences between the behaviors of the models with internal and external noises are also discussed.

Memory-induced resonancelike suppression of spike generation in a resonate-and-fire neuron model.

The behavior of a stochastic resonate-and-fire neuron model based on a reduction of a fractional noise-driven generalized Langevin equation (GLE) with a power-law memory kernel is considered. The effect of temporally correlated random activity of synaptic inputs, which arise from other neurons forming local and distant networks, is modeled as an additive fractional Gaussian noise in the GLE. Using a first-passage-time formulation, in certain system parameter domains exact expressions for the output interspike interval (ISI) density and for the survival probability (the probability that a spike is not generated) are derived and their dependence on input parameters, especially on the memory exponent, is analyzed. In the case of external white noise, it is shown that at intermediate values of the memory exponent the survival probability is significantly enhanced in comparison with the cases of strong and weak memory, which causes a resonancelike suppression of the probability of spike generation as a function of the memory exponent. Moreover, an examination of the dependence of multimodality in the ISI distribution on input parameters shows that there exists a critical memory exponent α_{c}≈0.402, which marks a dynamical transition in the behavior of the system. That phenomenon is illustrated by a phase diagram describing the emergence of three qualitatively different structures of the ISI distribution. Similarities and differences between the behavior of the model at internal and external noises are also discussed.

<em>L<sub>p</sub></em>-solutions of the Navier-Stokes equation with fractional Brownian noise

We study the Navier-Stokes equations on a smooth bounded domain $D\subset \mathbb R^d$ ($d = 2$ or 3), under the effect of an additive fractional Brownian noise. We show local existence and uniqueness of a mild $L^p$-solution for $p>d$.

Stochastic heat equation and martingale differences

ABSTRACTWe prove an approximation in law of the real-valued mild solution for a class of semilinear stochastic heat equation driven by an additive fractional noise using martingale differences under suitable assumptions.

Trichotomous noise induced stochastic resonance in a fractional oscillator with random damping and random frequency

.In this study, we investigate the stochastic resonance (SR) of a fractional linear oscillator subjected to multiplicative trichotomous noise and additive fractional Gaussian noise and driven by a periodic signal. Using the Shapiro-Loginov formula and the Laplace transformation technique, we acquire the exact expression of the first-order moment of the system’s steady response. Meanwhile, we discuss the evolutions of the output amplitude with the frequency of the periodic signal and noise parameters. We determine that bona fide SR, SR and reverse-resonance exist in this system. Specifically, the evolution of the output amplitude with the frequency of the periodic signal presents one-peak oscillation, double-peak oscillation and triple-peak oscillation. Moreover, the interplay of the trichotomous noise and memory can induce and diversify the stochastic multi-resonance (SMR) phenomena and give this linear system richer dynamic behavior. A hypersensitive response of the output amplitude to the noise intensity is demonstrated, which is not observed in systems driven by dichotomous noise.

Stochastic Resonance in a Fractional Oscillator with Random Mass and Random Frequency

For a fractional linear oscillator subjected to two multiplicative dichotomous noises and a additive fractional Gaussian noise and driven by a periodic signal, we study the stochastic resonance (SR) in this paper. Using (fractional) Shapiro–Loginov formula and the Laplace transformation technique, we acquire the exact expression of the first-order moment of the system’s steady response. Meanwhile, we discuss the evolutions of the output amplitude with frequency of the periodic signal, noise parameters, fractional order, and friction coefficient. We find that SR in the wide sense existing in this system. Specially, the evolution of the output amplitude with frequency of the periodic signal presents one-peak oscillation and two-peak oscillation. Moreover, the friction coefficient can induce stochastic multi-resonance.

Homogenization Driven by a Fractional Brownian Motion: The Shear Layer Case

We consider a passive scalar in a periodic shear flow perturbed by an additive fractional noise with the Hurst exponent $H\in(0,1)$. We establish a diffusive homogenization limit for the tracer when the Hurst exponent $H\in(0,1/2)$. We also identify an intermediate range of times when the tracer behaves diffusively even when $H\in(1/2,1)$. The proof is based on an auxiliary limit theorem for an additive functional of a fractional Brownian motion.

A least square-type procedure for parameter estimation in stochastic differential equations with additive fractional noise

We study a least square-type estimator for an unknown parameter in the drift coefficient of a stochastic differential equation with additive fractional noise of Hurst parameter \(H>1/2\). The estimator is based on discrete time observations of the stochastic differential equation, and using tools from ergodic theory and stochastic analysis we derive its strong consistency.

WEAK CONVERGENCE FOR QUASILINEAR STOCHASTIC HEAT EQUATION DRIVEN BY A FRACTIONAL NOISE WITH HURST PARAMETER H ∈ (½, 1)

In this paper, we consider a quasi-linear stochastic heat equation in one dimension on [0, 1], with Dirichlet boundary conditions driven by an additive fractional white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter n ∈ ℕ which can approximate the fractional noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space [Formula: see text] of continuous functions, to the solution of the fractional noise driven SPDE.

Weak approximation of the fractional Brownian sheet from random walks

In this paper, we show an approximation in law of the fractional Brownian sheet by random walks. As an application, we consider a quasilinear stochastic heat equation with Dirichlet boundary conditions driven by an additive fractional noise.

Stochastic Resonance in a Linear Fractional Langevin Equation

The fractional Langevin equation is derived from the generalized Langevin equation driven by the additive fractional Gaussian noise. We investigate the stochastic resonance (SR) phenomenon in the underdamped linear fractional Langevin equation under the external periodic force and multiplicative symmetric dichotomous noise. Applying the Shapiro-Loginov formula and the Laplace transform technique, we obtain the exact expressions of the amplitude and signal-to-noise ratio (SNR) of the system. By studying the impacts of the driving frequency and the noise parameters, we find the non-monotonic behaviors of the output amplitude and SNR. The results indicate that the bona fide SR, conventional SR and the wide sense of SR phenomena occur in the proposed linear fractional system.

Design for estimation of the drift parameter in fractional diffusion systems

We consider a controlled linear differential equation which is partially observed with an additive fractional noise. In this setting, we study the asymptotic (for large observation time) design problem of the input and give an efficient estimator of the unknown signal drift parameter. The optimal estimation input is deduced. The consistency, asymptotic normality and convergence of the moments of the MLE are established.

Apparent/spurious multifractality of absolute increments sampled from truncated fractional Gaussian/Lévy noise

[1] Many earth and environmental variables appear to scale as multifractals with spatial or temporal increments having exceedance probability tails decaying as powers of −α where 1 < α ≤ 2. The literature considers multifractal scaling to be associated with multiplicative random fields or processes. Elsewhere the author has demonstrated theoretically that square increments, sampled across a finite domain from one or several realizations of additive fractional Gaussian noise (fGn), behave as if the field was multifractal when in fact it is monofractal self-affine; square increments sampled from additive fractional Levy noise (fLn) with 1 < α < 2 exhibit spurious multifractality. This brief letter demonstrates the same numerically for random absolute increments. The results have broad implications vis-a-vis the scaling of variables considered in the literature to be multifractal, raising the possibility that some if not all may in fact represent truncated monofractal phenomena.

Multilevel Monte Carlo for stochastic differential equations with additive fractional noise

We adopt the multilevel Monte Carlo method introduced by M. Giles (Multilevel Monte Carlo path simulation, Oper. Res. 56(3):607–617, 2008) to SDEs with additive fractional noise of Hurst parameter H>1/2. For the approximation of a Lipschitz functional of the terminal state of the SDE we construct a multilevel estimator based on the Euler scheme. This estimator achieves a prescribed root mean square error of order e with a computational effort of order e−2.

Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion

In this article we study the behavior of dissipative systems with additive fractional noise of any Hurst parameter. Under a one-sided dissipative Lipschitz condition on the drift the continuous stochastic system is shown to have a unique stationary solution, which pathwise attracts all other solutions. The same holds for the discretized stochastic system, if the drift-implicit Euler method is used for the discretization. Moreover, the unique stationary solution of the drift-implicit Euler scheme converges to the unique stationary solution of the original system as the stepsize of the discretization decreases.

Mutual Information for Stochastic Signals and Fractional Brownian Motion

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <?Pub Dtl=""?>The mutual information between a stochastic signal and this signal plus a fractional Brownian motion (described as an additive fractional Gaussian noise channel) is expressed as the error of an estimation problem that can be naturally associated with this model. If the stochastic signal with the additive fractional Brownian motion occurs multiplied by a scalar parameter, then the rate of change of the mutual information with respect to this parameter is described by the error of another related estimation problem. These results generalize some results for a model where the fractional Brownian motion is a Brownian motion to a model with an arbitrary fractional Brownian motion. </para>

Optimal approximation of SDE's with additive fractional noise

We study pathwise approximation of scalar stochastic differential equations with additive fractional Brownian noise of Hurst parameter H > 1 2 , considering the mean square L 2 -error criterion. By means of the Malliavin calculus we derive the exact rate of convergence of the Euler scheme, also for non-equidistant discretizations. Moreover, we establish a sharp lower error bound that holds for arbitrary methods, which use a fixed number of bounded linear functionals of the driving fractional Brownian motion. The Euler scheme based on a discretization, which reflects the local smoothness properties of the equation, matches this lower error bound up to the factor 1.39.

Weak solutions for stochastic differential equations with additive fractional noise

In this paper, we show the existence of a weak solution for a stochastic differential equation driven by an additive fractional Brownian motion with Hurst parameter H > 1 2 , and a discontinuous drift. The proof of this result is based on the Girsanov theorem for the fractional Brownian motion.