scholarly journals Packing measure of the sample paths of fractional Brownian motion

1996 ◽  
Vol 348 (8) ◽  
pp. 3193-3213 ◽  
Author(s):  
Yimin Xiao
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Packing Measure ◽  
Sample Paths

scholarly journals A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

Generalized fractional Lévy processes with fractional Brownian motion limit

2015 ◽  
Vol 47 (04) ◽  
pp. 1108-1131 ◽  
Author(s):  
Claudia Klüppelberg ◽  
Muneya Matsui
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Lévy Processes ◽  
Sample Path ◽  
Levy Processes ◽  
Order Structure ◽  
Uhlenbeck Process ◽  
Sample Paths ◽  
Volatility Models ◽  
Ornstein Uhlenbeck Process

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernel to a regularly varying function. We call the resulting stochastic processes generalized fractional Lévy processes (GFLPs) and show that they may have short or long memory increments and that their sample paths may have jumps or not. Moreover, we define stochastic integrals with respect to a GFLP and investigate their second-order structure and sample path properties. A specific example is the Ornstein-Uhlenbeck process driven by a time-scaled GFLP. We prove a functional central limit theorem for such scaled processes with a fractional Ornstein-Uhlenbeck process as a limit process. This approximation applies to a wide class of stochastic volatility models, which include models where possibly neither the data nor the latent volatility process are semimartingales.

scholarly journals On the mixed fractional Brownian motion

2006 ◽  
Vol 2006 ◽  
pp. 1-9 ◽  
Author(s):  
Mounir Zili
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Mathematical Finance ◽  
Complete Markets ◽  
Sample Paths ◽  
Mixed Fractional Brownian Motion ◽  
Stochastic Properties

The mixed fractional Brownian motion is used in mathematical finance, in the modelling of some arbitrage-free and complete markets. In this paper, we present some stochastic properties and characteristics of this process, and we study the α-differentiability of its sample paths.

scholarly journals Sample Paths Estimates for Stochastic Fast-Slow Systems Driven by Fractional Brownian Motion

2020 ◽  
Vol 179 (5-6) ◽  
pp. 1222-1266 ◽  
Author(s):  
Katharina Eichinger ◽  
Christian Kuehn ◽  
Alexandra Neamţu
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Sample Paths

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.

A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm

2014 ◽  
Vol 51 (01) ◽  
pp. 1-18 ◽  
Author(s):  
Dawei Hong ◽  
Shushuang Man ◽  
Jean-Camille Birget ◽  
Desmond S. Lun
Keyword(s):  
Brownian Motion ◽  
Convergence Rate ◽  
Fractional Brownian Motion ◽  
Parallel Algorithm ◽  
Uniform Approximation ◽  
Haar Wavelets ◽  
Hurst Index ◽  
Sample Paths ◽  
Vanishing Moment ◽  
Uniform Expansion

We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (Bt(H))_t∈[0,1]of Hurst indexH∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM forH∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.

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.

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