scholarly journals On the Self-Intersection Local Time of Subfractional Brownian Motion

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Junfeng Liu ◽  
Zhihang Peng ◽  
Donglei Tang ◽  
Yuquan Cang
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Local Time ◽  
Noise Analysis ◽  
The Self ◽  
White Noise Analysis ◽  
Intersection Local Time ◽  
Subfractional Brownian Motion

We study the problem of self-intersection local time ofd-dimensional subfractional Brownian motion based on the property of chaotic representation and the white noise analysis.

scholarly journals The 𝒮-Transform of Sub-fBm and an Application to a Class of Linear Subfractional BSDEs

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Zhi Wang ◽  
Litan Yan
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
White Noise ◽  
Stochastic Differential Equations ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Explicit Solutions ◽  
Pettis Integral ◽  
White Noise Analysis ◽  
Subfractional Brownian Motion

Let SH be a subfractional Brownian motion with index 0<H<1. Based on the 𝒮-transform in white noise analysis we study the stochastic integral with respect to SH, and we also prove a Girsanov theorem and derive an Itô formula. As an application we study the solutions of backward stochastic differential equations driven by SH of the form -dYt=f(t,Yt,Zt)dt-ZtdStH, t∈[0,T],YT=ξ, where the stochastic integral used in the above equation is Pettis integral. We obtain the explicit solutions of this class of equations under suitable assumptions.

White-noise analysis in visual neuroscience

1988 ◽  
Vol 1 (3) ◽  
pp. 287-296 ◽  
Author(s):  
Hiroko M. Sakai ◽  
Naka Ken-Ichi ◽  
Michael J. Korenberg
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Light Stimulus ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
White Noise Analysis ◽  
Response Dynamics ◽  
Visual Neuroscience ◽  
Final Output ◽  
Mean Luminance

AbstractIn 1827, plant biologist Robert Brown discovered what is known as Brownian motion, a class of chaos. Formal derivative of Brownian motion is Gaussian white-noise. In 1938, Norbert Wiener proposed to use the Gaussian white-noise as an input probe to identify a system by a series of orthogonal functionals known as the Wiener G-functionals.White-noise analysis is uniquely suited for studying the response dynamics of retinal neurons because (1) white-noise light stimulus is a modulation around a mean luminance, as are the natural photic inputs, and it is a highly efficient input; and (2) the analysis defines the response dynamics and can be extended to spike trains, the final output of the retina. Demonstrated here are typical examples and results from applications of white-noise analysis to a visual system.

scholarly journals Local times of self-intersection for multidimensional Brownian motion

1995 ◽  
Vol 138 ◽  
pp. 51-64 ◽  
Author(s):  
Sheng-Wu He ◽  
Wen-Qiang Yang ◽  
Rong-Qin Yao ◽  
Jia-Gang Wang
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Main Part ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Deep Insight ◽  
Insight Into

We will define local times of self-intersection for multidimensional Brownian motion as generalized Wiener functionals under the framework of white noise analysis as in H. Watanabe ([6]). By making use of the chaotic representation of -function and precise computation we get a deep insight into the problem. In the section 1 multiple Wiener integrals with respect to multidimensional Brownian motion and chaotic representations for square-integrable Wiener functionals are given. They are indispensable, but seem not to be formulated clearly and correctly before. The useful concepts and results of white noise analysis are illustrated in the section 2. Section 3 is the main part of the paper. The applications to local times are introduced in the section 4 briefly.

Self-intersection local times for multifractional Brownian motion in higher dimensions: A white noise approach

2020 ◽  
Vol 23 (01) ◽  
pp. 2050007
Author(s):  
Wolfgang Bock ◽  
Jose Luis da Silva ◽  
Herry Pribawanto Suryawan
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Noise Analysis ◽  
Higher Dimensions ◽  
Kernel Functions ◽  
The Self ◽  
Local Times ◽  
Multifractional Brownian Motion ◽  
White Noises ◽  
Intersection Local Times

In this paper, we study the self-intersection local times of multifractional Brownian motion (mBm) in higher dimensions in the framework of white noise analysis. We show that when a suitable number of kernel functions of self-intersection local times of mBm are truncated then we obtain a Hida distribution. In addition, we present the expansion of the self-intersection local times in terms of Wick powers of white noises. Moreover, we obtain the convergence of the regularized truncated self-intersection local times in the sense of Hida distributions.

scholarly journals Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

2019 ◽  
Vol 15 (2) ◽  
pp. 81 ◽  
Author(s):  
Herry Pribawanto Suryawan
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Short Range ◽  
Probability Space ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Self Similarity ◽  
Sample Paths

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

scholarly journals White noise analysis and Tanaka formula for intersections of planar Brownian motion

1991 ◽  
Vol 122 ◽  
pp. 1-17 ◽  
Author(s):  
Narn-Rueih Shieh
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Stochastic Integral ◽  
Integral Formula ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Planar Brownian Motion ◽  
Tanaka Formula ◽  
Intersection Local Times

In this paper, we shall use Hida’s [5, 7, 9] theory of generalized Brownian functionals (or named white noise analysis) to establish a stochastic integral formula concerning the multiple intersection local times of planar Brownian motion B(t).

scholarly journals Weighted Local Times of a Sub-fractional Brownian Motion as Hida Distributions

2020 ◽  
Vol 15 (2) ◽  
pp. 81
Author(s):  
Herry Pribawanto Suryawan
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Fractional Brownian Motion ◽  
Short Range ◽  
Probability Space ◽  
Noise Analysis ◽  
Local Times ◽  
White Noise Analysis ◽  
Self Similarity ◽  
Sample Paths

The sub-fractional Brownian motion is a Gaussian extension of the Brownian motion. It has the properties of self-similarity, continuity of the sample paths, and short-range dependence, among others. The increments of sub-fractional Brownian motion is neither independent nor stationary. In this paper we study the sub-fractional Brownian motion using a white noise analysis approach. We recall the represention of sub-fractional Brownian motion on the white noise probability space and show that Donsker's delta functional of a sub-fractional Brownian motion is a Hida distribution. As a main result, we prove the existence of the weighted local times of a $d$-dimensional sub-fractional Brownian motion as Hida distributions.

scholarly journals NO ZERO DIVISOR FOR WICK PRODUCT IN (S)*

2008 ◽  
Vol 11 (02) ◽  
pp. 307-311 ◽  
Author(s):  
TAKAHIRO HASEBE ◽  
IZUMI OJIMA ◽  
HAYATO SAIGO
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Zero Divisor ◽  
Noise Analysis ◽  
Operational Calculus ◽  
Generalized Functions ◽  
Wick Product ◽  
White Noise Analysis ◽  
Remarkable Property ◽  
Zero Divisors

In White Noise Analysis (WNA), various random quantities are analyzed as elements of (S)*, the space of Hida distributions.1 Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On (S)*, the Wick product is defined in terms of the [Formula: see text]-transform. We have found such a remarkable property that the Wick product has no zero divisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the "operational calculus" in WNA along the line of Mikusiński's version for solving differential equations.

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