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 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.

scholarly journals Existence and exponential stability in the pth moment for impulsive neutral stochastic integro-differential equations driven by mixed fractional Brownian motion

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Xia Zhou ◽  
Dongpeng Zhou ◽  
Shouming Zhong
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Exponential Stability ◽  
Fractional Brownian Motion ◽  
Mild Solutions ◽  
Semigroup Theory ◽  
Sufficient Conditions ◽  
Fractional Power ◽  
Banach Fixed Point Theorem ◽  
Mixed Fractional Brownian Motion

Abstract This paper consider the existence, uniqueness and exponential stability in the pth moment of mild solution for impulsive neutral stochastic integro-differential equations driven simultaneously by fractional Brownian motion and by standard Brownian motion. Based on semigroup theory, the sufficient conditions to ensure the existence and uniqueness of mild solutions are obtained in terms of fractional power of operators and Banach fixed point theorem. Moreover, the pth moment exponential stability conditions of the equation are obtained by means of an impulsive integral inequality. Finally, an example is presented to illustrate the effectiveness of the obtained results.

scholarly journals Solving Arbitrage Problem on the Financial Market Under the Mixed Fractional Brownian Motion With Hurst Parameter H ∈]1/2,3/4[

2019 ◽  
Vol 11 (1) ◽  
pp. 76
Author(s):  
Eric Djeutcha ◽  
Didier Alain Njamen Njomen ◽  
Louis-Aimé Fono
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Financial Market ◽  
Existence And Uniqueness ◽  
Hurst Parameter ◽  
Existence And Uniqueness Theorem ◽  
Martingale Approximation ◽  
Arbitrage Opportunities ◽  
Mixed Fractional Brownian Motion ◽  
Geometric Fractional Brownian Motion

This study deals with the arbitrage problem on the financial market when the underlying asset follows a mixed fractional Brownian motion. We prove the existence and uniqueness theorem for the mixed geometric fractional Brownian motion equation. The semi-martingale approximation approach to mixed fractional Brownian motion is used to eliminate the arbitrage opportunities.

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