Approximation of Fractional Brownian Motion by Martingales

Fractional martingales and characterization of the fractional Brownian motion

In this paper we introduce the notion of α-martingale as the fractional derivative of order α of a continuous local martingale, where α ∈ (−12 , 1 2), and we show that it has a nonzero finite variation of order 2 1+2α , under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy’s characterization theorem for the f...

Fractional Brownian fields, duality, and martingales

In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian moti...

Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion

Semimartingale approximation of fractional Brownian motion and its applications

The aim of this paper is to provide a semimartingale approximation of a fractional stochastic integration. This result leads us to approximate the fractional Black-Scholes model by a model driven by semimartingales, and a European option pricing formula is found. 2000 AMS Classification: 60H05, 65G15, 62P05.

A weighted random walk approximation to fractional Brownian motion

The purpose of this brief note is to describe a discrete approximation to fractional Brownian motion. The approximation works for all Hurst indices H , but take slightly different forms for H ≤ 1 2 and H > 1 2 . There are already several discrete approximations to fractional Brownian motion in the literature (see, e.g., [11], [1], [3], [10], [4], [2], [5], [8] for this and related topics), and ...