Time Of Fractional Brownian Motion Research Articles

Higher-order derivative of local times for space–time anisotropic Gaussian random fields

Let X={X(t),t∈RN} be a centered space–time anisotropic Gaussian random field values in Rd. Under some general conditions, the existence and smoothness (in the sense of Meyer-Watanabe) of the higher-order derivative of the local times of X(t) are studied. Moreover, we show that the derivatives of the local time of X(t) is jointly continuous on Rd×[0,1]N. The existing results on local times of fractional Brownian motion and other Gaussian random fields are extended to higher-order derivative of local times of more general space–time anisotropic Gaussian random fields.

The First Exit Time of Fractional Brownian Motion with a Drift from a Parabolic Domain

The First Exit Time of Fractional Brownian Motion with a Drift from a Parabolic Domain

Stochastic ordering for hitting times of fractional Brownian motions

In this note we prove that, under a suitable transformation, the hitting times of fractional Brownian motions (fBMs) can be ordered stochastically. As a consequence, we derive upper and lower bounds for their cumulative distribution function. We also study some properties, such as continuity and monotonicity, of the probability tail and of the expectation of the supremum of fBMs seen as functions of the Hurst index H. We apply these properties to provide alternative proofs of some results known for fBM.

An extension of the stochastic sewing lemma and applications to fractional stochastic calculus

Abstract We give an extension of Lê’s stochastic sewing lemma. The stochastic sewing lemma proves convergence in $L_m$ of Riemann type sums $\sum _{[s,t] \in \pi } A_{s,t}$ for an adapted two-parameter stochastic process A, under certain conditions on the moments of $A_{s,t}$ and of conditional expectations of $A_{s,t}$ given $\mathcal F_s$ . Our extension replaces the conditional expectation given $\mathcal F_s$ by that given $\mathcal F_v$ for $v<s$ , and it allows to make use of asymptotic decorrelation properties between $A_{s,t}$ and $\mathcal F_v$ by including a singularity in $(s-v)$ . We provide three applications for which Lê’s stochastic sewing lemma seems to be insufficient. The first is to prove the convergence of Itô or Stratonovich approximations of stochastic integrals along fractional Brownian motions under low regularity assumptions. The second is to obtain new representations of local times of fractional Brownian motions via discretization. The third is to improve a regularity assumption on the diffusion coefficient of a stochastic differential equation driven by a fractional Brownian motion for pathwise uniqueness and strong existence.

Derivative of Multiple Self-intersection Local Time for Fractional Brownian Motion

Derivative of Multiple Self-intersection Local Time for Fractional Brownian Motion

Poincaré Plot Nonextensive Distribution Entropy: A New Method for Electroencephalography (EEG) Time Series.

As a novel form of visual analysis technique, the Poincaré plot has been used to identify correlation patterns in time series that cannot be detected using traditional analysis methods. In this work, based on the nonextensive of EEG, Poincaré plot nonextensive distribution entropy (NDE) is proposed to solve the problem of insufficient discrimination ability of Poincaré plot distribution entropy (DE) in analyzing fractional Brownian motion time series with different Hurst indices. More specifically, firstly, the reasons for the failure of Poincaré plot DE in the analysis of fractional Brownian motion are analyzed; secondly, in view of the nonextensive of EEG, a nonextensive parameter, the distance between sector ring subintervals from the original point, is introduced to highlight the different roles of each sector ring subinterval in the system. To demonstrate the usefulness of this method, the simulated time series of the fractional Brownian motion with different Hurst indices were analyzed using Poincaré plot NDE, and the process of determining the relevant parameters was further explained. Furthermore, the published sleep EEG dataset was analyzed, and the results showed that the Poincaré plot NDE can effectively reflect different sleep stages. The obtained results for the two classes of time series demonstrate that the Poincaré plot NDE provides a prospective tool for single-channel EEG time series analysis.

The first exit time of fractional Brownian motion from the minimum and maximum parabolic domains

Consider a fractional Brownian motion starting at an interior point of the minimum and maximum parabolic domains, namely, Dmin=x,y1,y2:‖x‖<minj=1,2aj+yj1/pj and Dmax=x,y1,y2:‖x‖<maxj=1,2aj+yj1/pj in Rd+2, d≥1, where ‖⋅‖ is the Euclidean norm in Rd, a1,a2>0, and p1,p2>1. Let τmin and τmax denote the first times that the fractional Brownian motion exits from Dmin and Dmax, respectively. Asymptotically equivalent estimates of logPτmin>t and logPτmax>t are respectively given by using Gordon’s inequality, depending on the relationship between p1 and p2. The proof methods are based on early works of Li, Shi, Lifshits, Aurzada and Lu.

Regularization by random translation of potentials for the continuous PAM and related models in arbitrary dimension

We study a regularization by noise phenomenon for the continuous parabolic Anderson model with a potential shifted along paths of fractional Brownian motion. We demonstrate that provided the Hurst parameter is chosen sufficiently small, this shift allows to establish well-posedness and stability to the corresponding problem – without the need of renormalization – in any dimension. We moreover provide a robustified Feynman-Kac type formula for the unique solution to the regularized problem building upon regularity estimates for the local time of fractional Brownian motion as well as non-linear Young integration.

Smoothness of higher order derivative of self-intersection local time for fractional Brownian motion

In a recent paper, Yu (2021) define a -th order derivative of self-intersection local times with respect to d-dimensional fractional Brownian motion. By the existence and Hölder continuity proved in Yu (2021), we study the smoothness of derivative of self-intersection local times with respect to fractional Brownian motion in this paper. We also consider the cases of intersection local times and collision local times, respectively.

A computational technique to classify several fractional Brownian motion processes

In this paper, for the first time, the classification of several fractional Brownian motion time series is considered. For this purpose, fuzzy clustering technique is applied and Brownian motion processes are classified. The applicability of the given approach is explored using simulated and a real COVID-19 dataset.

Higher-Order Derivative of Self-Intersection Local Time for Fractional Brownian Motion

We consider the existence and Hölder continuity conditions for the k-th-order derivatives of self-intersection local time for d-dimensional fractional Brownian motion, where $$k=(k_1,k_2,\ldots , k_d)$$ . Moreover, we show a limit theorem for the critical case with $$H=\frac{2}{3}$$ and $$d=1$$ , which was conjectured by Jung and Markowsky [7].

Existence, renormalization, and regularity properties of higher order derivatives of self-intersection local time of fractional Brownian motion

In a recent paper by Yu (arXiv:2008.05633, 2020), higher order derivatives of self-intersection local time of fractional Brownian motion were defined, and existence over certain regions of the Hurst parameter H was proved. Utilizing the Wiener chaos expansion, we provide new proofs of Yu’s results and show how a Varadhan-type renormalization can be used to extend the range of convergence for the even derivatives.

Asymptotic properties for q-th chaotic component of derivative of self-intersection local time of fractional Brownian motion

Let {BtH}t≥0 be a fractional Brownian motion with Hurst parameter H∈(0,1). Consider the approximation of derivative of self-intersection local time of BH, defined asαε=∫0T∫0tpε′(BtH−BsH)dsdt, where pε(x) is the heat kernel. For q≥2, renormalized by different factors, we prove two different central limit theorems for q-th chaotic component of αε when H=34 and H=4q−34q−2, respectively. The results are obtained based on the fourth moment theorem introduced in [8].

On the mixed fractional Brownian motion time changed by inverse α-stable subordinator

On the mixed fractional Brownian motion time changed by inverse α-stable subordinator

The Csörgő–Révész moduli of non-differentiability of fractional Brownian motion

We establish the exact moduli of non-differentiability of fractional Brownian motion. As an application of the result, we prove that the uniform Hölder condition for the maximum local times of fractional Brownian motion obtained in Xiao (1997) is optimal.

The First Exit Time of Fractional Brownian Motion from a Parabolic Domain

We study the first exit time of a multi-dimensional fractional Brownian motion from unbounded domains. In particular, we are interested in the upper tail of the corresponding distribution when the domain is parabola-shaped.

The first exit time of fractional Brownian motion from a parabolic domain

Изучается момент первого выхода многомерного дробного броуновского движения из неограниченных областей. В частности, исследуется верхний хвост соответствующего распределения в случае, когда область имеет форму параболы.

Renormalized self-intersection local time of bifractional Brownian motion

Let B^{H,K}={B^{H,K}(t), t geq 0} be a d-dimensional bifractional Brownian motion with Hurst parameters Hin (0,1) and Kin (0,1]. Assuming dgeq 2, we prove that the renormalized self-intersection local time \t\t\t∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt−E(∫0T∫0tδ(BH,K(t)−BH,K(s))dsdt)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} \\int^{T}_{0} \\int^{t}_{0}\\delta \\bigl(B^{H,K}(t)-B^{H,K}(s) \\bigr)\\,ds\\,dt-\\mathbb{E} \\biggl( \\int^{T}_{0} \\int^{t}_{0}\\delta \\bigl(B^{H,K}(t)-B^{H,K}(s) \\bigr)\\,ds\\,dt \\biggr) \\end{aligned}$$ \\end{document} exists in L^{2} if and only if HKd< 3/2, where δ denotes the Dirac delta function. Our work generalizes the result of the renormalized self-intersection local time for fractional Brownian motion.

Evenly spaced Detrended Fluctuation Analysis: Selecting the number of points for the diffusion plot

Detrended Fluctuation Analysis (DFA) has become a widely-used tool to examine the correlation structure of a time series and provided insights into neuromuscular health and disease states. As the popularity of utilizing DFA in the human behavioral sciences has grown, understanding its limitations and how to properly determine parameters is becoming increasingly important. DFA examines the correlation structure of variability in a time series by computing α, the slope of the logSD–logn diffusion plot. When using the traditional DFA algorithm, the timescales, n, are often selected as a set of integers between a minimum and maximum length based on the number of data points in the time series. This produces non-uniformly distributed values of n in logarithmic scale, which influences the estimation of α due to a disproportionate weighting of the long-timescale regions of the diffusion plot. Recently, the evenly spaced DFA and evenly spaced average DFA algorithms were introduced. Both algorithms compute α by selecting k points for the diffusion plot based on the minimum and maximum timescales of interest and improve the consistency of α estimates for simulated fractional Gaussian noise and fractional Brownian motion time series. Two issues that remain unaddressed are (1) how to select k and (2) whether the evenly-spaced DFA algorithms show similar benefits when assessing human behavioral data. We manipulated k and examined its effects on the accuracy, consistency, and confidence limits of α in simulated and experimental time series. We demonstrate that the accuracy and consistency of α are relatively unaffected by the selection of k. However, the confidence limits of α narrow as k increases, dramatically reducing measurement uncertainty for single trials. We provide guidelines for selecting k and discuss potential uses of the evenly spaced DFA algorithms when assessing human behavioral data.

Representation of local times of fractional Brownian motion

In this paper we obtain a simple representation of the classical local times of fractional Brownian motion. We explore the properties of the quadratic variation of fractional Brownian motion {X(t):t≥0} of index 0<H≤1∕2 and obtain a generalisation of Tanaka’s formula from which we deduce that the local times at any point a are given by L(t,a)=limn→∞2n(2H−1)∑k∈Sℓ2|X(k2−n)−a|with Sℓ={k∈{1,2,…,ℓ}:(X(k2−n)−a)(X((k−1)2−n)−a)<0} and ℓ=⌊t2n⌋. This is the simplest expression of local times known to the author and it is amenable to numerical computations. Our arguments are elementary and do not use stochastic integration with respect to fractional Brownian motion.