Increments Of Fractional Brownian Motion Research Articles

Real zeros of random trigonometric polynomials with dependent coefficients

We further investigate the relations between the large degree asymptotics of the number of real zeros of random trigonometric polynomials with dependent coefficients and the underlying correlation function. We consider trigonometric polynomials of the form \[ f n ( t ) ≔ 1 n ∑ k = 1 n a k cos ⁡ ( k t ) + b k sin ⁡ ( k t ) , x ∈ [ 0 , 2 π ] , f_n(t)≔\frac {1}{\sqrt {n}}\sum _{k=1}^{n}a_k \cos (kt)+b_k\sin (kt), ~x\in [0,2\pi ], \] where the sequences ( a k ) k ≥ 1 (a_k)_{k\geq 1} and ( b k ) k ≥ 1 (b_k)_{k\geq 1} are two independent copies of a stationary Gaussian process centered with variance one and correlation function ρ \rho with associated spectral measure μ ρ \mu _{\rho } . We focus here on the case where μ ρ \mu _{\rho } is not purely singular and we denote by ψ ρ \psi _{\rho } its density component with respect to the Lebesgue measure λ \lambda . Quite surprisingly, we show that the asymptotics of the number of real zeros N ( f n , [ 0 , 2 π ] ) \mathcal {N}(f_n,[0,2\pi ]) of f n f_n in [ 0 , 2 π ] [0,2\pi ] is not related to the decay of the correlation function ρ \rho but instead to the Lebesgue measure of the vanishing locus of ψ ρ \psi _{\rho } . Namely, assuming that ψ ρ \psi _{\rho } is C 1 \mathcal {C}^1 with Hölder derivative on an open set of full measure, one establishes that \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = λ ( { ψ ρ = 0 } ) π 2 + 2 π − λ ( { ψ ρ = 0 } ) π 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {\lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {2}} + \frac {2\pi - \lambda (\{\psi _{\rho }=0\})}{\pi \sqrt {3}}. \] On the other hand, assuming a sole log-integrability condition on ψ ρ \psi _{\rho } , which implies that it is positive almost everywhere, we recover the asymptotics of the independent case: \[ lim n → + ∞ E [ N ( f n , [ 0 , 2 π ] ) ] n = 2 3 . \lim _{n \to +\infty } \frac {\mathbb {E}\left [\mathcal {N}(f_n,[0,2\pi ])\right ]}{n}= \frac {2}{\sqrt {3}}. \] The latter asymptotics thus broadly generalizes the main result of Angst, Dalmao, and Poly [Proc. Amer. Math. Soc. 147 (2019), pp. 205–214] where the spectral density was assumed to be continuous and bounded from below. Besides, with further assumptions of regularity and existence of negative moment for ψ ρ \psi _{\rho } , which encompass e.g. the case of random coefficients being increments of fractional Brownian motion with any Hurst parameter, we moreover show that the above convergence in expectation can be strengthened to an almost sure convergence.

Tempered fractional Brownian motion on finite intervals

Diffusive transport in many complex systems features a crossover between anomalous diffusion at short times and normal diffusion at long times. This behavior can be mathematically modeled by cutting off (tempering) beyond a mesoscopic correlation time the power-law correlations between the increments of fractional Brownian motion. Here, we investigate such tempered fractional Brownian motion confined to a finite interval by reflecting walls. Specifically, we explore how the tempering of the long-time correlations affects the strong accumulation and depletion of particles near reflecting boundaries recently discovered for untempered fractional Brownian motion. We find that exponential tempering introduces a characteristic size for the accumulation and depletion zones but does not affect the functional form of the probability density close to the wall. In contrast, power-law tempering leads to more complex behavior that differs between the superdiffusive and subdiffusive cases.

High-frequency trading with fractional Brownian motion

In the high-frequency limit, conditionally expected increments of fractional Brownian motion converge to a white noise, shedding their dependence on the path history and the forecasting horizon and making dynamic optimisation problems tractable. We find an explicit formula for locally mean–variance optimal strategies and their performance for an asset price that follows fractional Brownian motion. Without trading costs, risk-adjusted profits are linear in the trading horizon and rise asymmetrically as the Hurst exponent departs from Brownian motion, remaining finite as the exponent reaches zero while diverging as it approaches one. Trading costs penalise numerous portfolio updates from short-lived signals, leading to a finite trading frequency, which can be chosen so that the effect of trading costs is arbitrarily small, depending on the required speed of convergence to the high-frequency limit.

High-Frequency Trading with Fractional Brownian Motion

In the high-frequency limit, conditional expected increments of fractional Brownian motion converge to a white noise, shedding their dependence on the path history and the forecasting horizon, and making dynamic optimization problems tractable. We find an explicit formula for locally mean-variance optimal strategies and their performance for an asset price that follows fractional Brownian motion. Without trading costs, risk-adjusted profits are linear in the trading horizon and rise asymmetrically as the Hurst exponent departs from Brownian motion, remaining finite as the exponent reaches zero while diverging as it approaches one. Trading costs penalize numerous portfolio updates from short-lived signals, leading to a finite trading frequency, which can be chosen so that the effect of trading costs is arbitrarily small, depending on the required speed of convergence to the high-frequency limit.

Almost sure limit theorems on Wiener chaos: the non-central case

In [1], a framework to prove almost sure central limit theorems for sequences $(G_n)$ belonging to the Wiener space was developed, with a particular emphasis of the case where $G_n$ takes the form of a multiple Wiener-Ito integral with respect to a given isonormal Gaussian process. In the present paper, we complement the study initiated in [1], by considering the more general situation where the sequence $(G_n)$ may not need to converge to a Gaussian distribution. As an application, we prove that partial sums of Hermite polynomials of increments of fractional Brownian motion satisfy an almost sure limit theorem in the long-range dependence case, a problem left open in [1].

Average Number of Real Zeros of Random Algebraic Polynomials Defined by the Increments of Fractional Brownian Motion

The study of random polynomials has a long and rich history. This paper studies random algebraic polynomials $$P_n(x) = a_0 + a_1 x + \cdots + a_{n-1} x^{n-1}$$ where the coefficients $$(a_k)$$ are correlated random variables taken as the increments $$X(k+1) - X(k)$$ , $$k\in \mathbb {N}$$ , of a fractional Brownian motion X of Hurst index $$0< H < 1$$ . This reduces to the classical setting of independent coefficients for $$H = 1/2$$ . We obtain that the average number of the real zeros of $$P_n(x)$$ is $$\sim K_H \log n$$ , for large n, where $$K_H = (1 + 2 \sqrt{H(1-H)})/\pi $$ [a generalisation of a classical result obtained by Kac (Bull Am Math Soc 49:314–320, 1943)]. Unexpectedly, the parameter H affects only the number of positive zeros, and the number of real zeros of the polynomials corresponding to fractional Brownian motions of indexes H and $$1-H$$ is essentially the same. The limit case $$H = 0$$ presents some particularities: the average number of positive zeros converges to a constant. These results shed some light on the nature of fractional Brownian motion, on the one hand, and on the behaviour of real zeros of random polynomials of dependent coefficients, on the other hand.

Quasi sure large deviation for increments of fractional Brownian motion in Hölder norm

In this paper, we first prove Schilder's theorem in Holder norm (0 ≤ α< 1) with respect to Cr,p-capacity. Then, based on this result, we further prove a sharpening of large deviation principle for increments of fractional Brownian motion for Cr,p-capacity in the stronger topology.

Using Fractional Gaussian Noise Models in Orbit Determination

Motivated by a recent requirement to provide accurate covariance matrices as well as orbit estimates for cata-loging space objects, a Bayesian estimator to handle autocorrelated process and measurement noises is presented. Although the application is orbit determination, the estimation method itself is general. Techniques are presented to model autocorrelated noises in terms of fractional Gaussian noise and increments of fractional Brownian motion. The Bayesian estimator is derived in both batch and sequential forms, along with explicit formulas for calculating the estimation error covariance matrix. The sequential form generalizes the Kalman filter to the case of autocorrelated noise processes. Also discussed is a batch least-squares estimator formulated with a whitening transformation that accounts for both measurement and process noise. Basic properties of the fractional Gaussian noise model are presented, showing how it contains the special case of white noise in a one-parameter family of self-similar random processes. The parameter characterizes the extent of the autocorrelation and is known as the Hurst parameter. When the measurement noise can be isolated from the process noise by appropriate sensor calibrations, statistical test-of-hypothesis techniques can be used to estimate the Hurst parameter by adjusting it to optimize thep value of the test. For a measurement noise sample that is sufficiently dense in time, the Hurst parameter can be calculated directly from the sample's fractal dimension. In contrast, the process noise usually cannot be isolated from the total noise because the measurements are the only information available. However, assuming the Hurst parameter of the measurement noise is either estimated first or known a priori, test-of-hypothesis techniques can be used to estimate the Hurst parameters of the process noise.

On Upper Class of Increments of Fractional Brownian Motion

A series test for the upper upper class and lower upper class of increments of fractional brownian motion is provided.

Improvements in Grade Tonnage Curve Prediction via Sequential Gaussian Fractal Simulation

An application of sequential Gaussian fractal conditional simulation is presented, using actual sparse, irregularly spaced open cut gold data, to predict the grade tonnage curve associated with a single mining bench. It is shown that this constitutes an improvement over the prediction of the grade tonnage curves obtained via standard sequential Gaussian simulation and median indicator simulation. The fractal conditional simulation method uses the model of the covariance of increments of fractional Brownian motion together with the fractal co-dimension derived from a truncated power semivariogram model to create conditional simulations from irregularly spaced data. The grades and tonnages above several discrete cut offs are calculated for each of one hundred simulations, and the mean and variance of the grade and tonnage values for each cut off are computed to give an average grade tonnage curve with associated confidence limits.

On the large increments of fractional Brownian motion

Let { B H ( t), t ⩾ 0} be a fractional Brownian motion with index 0 < H < 1 and define the statistic V T = sup 0⩽s⩽T−a T β T|B H(s + a T) − B H(s)| , where β T and α T are suitably chosen functions of T ⩾ 0. We establish some laws of the iterated logarithm for V T .

Some liminf results on increments of fractional Brownian motion

Some liminf results on increments of fractional Brownian motion

Fractal estimation from noisy data via discrete fractional Gaussian noise (DFGN) and the Haar basis

The authors show that the application of the discrete wavelet transform (DWT) using the Haar basis to the increments of fractional Brownian motion (fBm), also known as discrete fractional Gaussian noise (DFGN), yields coefficients which are weakly correlated and have a variance that is exponentially related to scale. Similar results were derived by Flandrin (1989), Tewfik, and Kim for a continuous-time fBm going through a continuous wavelet transform (CWT). The new theoretical results justify an improvement to a fractal estimation algorithm recently proposed by Wornell and Oppenheim. The performance of the new algorithm is compared with that of Wornell and Oppenheim's (see IEEE Trans. Signal Processing, vol. 40, p. 611-623, Mar. 1992) algorithm in numerical simulation. >