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2014 | OriginalPaper | Chapter

2. Preliminaries

Authors : Corinne Berzin, Alain Latour, José R. León

Published in: Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion

Publisher: Springer International Publishing

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Abstract

The chapter introduces the basic tools we use. First the fBm, its harmonizable representation and some of its properties are given, as well as an evaluation of its modulus of continuity. Also it is shown that fBm is not a semi-martingale. Then some preliminaries about stochastic integration with respect to the fBm are introduced. The attention will be focused on pathwise integrals. The existence of the integral as the limit in probability of the Riemann sums is proved. Afterwards, the notion of a stochastic differential equation driven by fractional noise is given, explaining next how one can construct the solution of such an equation. Moreover, it is established that the definitions of backwards, forwards and symmetric pathwise integrals coincide for some class of integrant functions. After the complex Itô-Wiener Chaos is defined and some tools useful for the subject are provided: Mehler’s formula, normalized second order increments and its covariance function. The chapter concludes by defining generalized variations.

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Metadata
Title
Preliminaries
Authors
Corinne Berzin
Alain Latour
José R. León
Copyright Year
2014
Book
Inference on the Hurst Parameter and the Variance of Diffusions Driven by Fractional Brownian Motion

Print ISBN: 978-3-319-07874-8

Electronic ISBN: 978-3-319-07875-5

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-07875-5_2

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