Stochastic calculus for fractional Brownian motion and related processes.
type: Item
instance of: scholarly article
title: Stochastic calculus for fractional Brownian motion and related processes.
zbMATH Open document ID: 1138.60006
DOI: 10.1007/978-3-540-75873-0
published in: Lecture Notes in Mathematics
publication date: 2007-12-06T00:00:00Z
review text: The volume addresses the general theory of fractional Brownian motion and other related stochastic processes with long-memory. It also contains applications of the theory for solving the problems arising in optimal filtering, financial mathematics and statistical inference for models involving fractional Brownian motions. The book consists of six chapters and is organized as follows. Chapter 1 is devoted to the integration of non-random integrands with respect to a fractional Brownian motion. It recalls, in particular, representations of a fractional Brownian motion as an integral of a fractional kernel with respect to a Wiener process and of some other auxiliary processes, Wick calculus for mixed Brownian-fractional Brownian motion, stochastic Fubini's theorem for the integrals with respect to fractional Brownian motions, different approximation schemes for fractional Brownian motions using semimartingales and continuous processes of bounded variation, an analogue of Lévy characterization theorem for a fractional Brownian motion, and Wiener fields on the plane. Chapter 2 deals with the stochastic integration with respect to fractional Brownian motion and other aspects of the related stochastic calculus. The considered different approaches include pathwise integration, Wick integration, Skorohod integration and isometric integration. The related stochastic Fubini's theorem, various versions of Itô's formula and Girsanov's theorem are presented. Chapter 3 provides a study of different properties of stochastic differential equations driven by fractional Brownian motion. It contains, in particular, the existence and uniqueness of local and global solutions of such stochastic differential equations, the estimates for the solutions, and the related Euler approximation schemes with its convergence. Chapter 4 is devoted to optimal filtering problems in the mixed Brownian-fractional Brownian models. It contains the case where the signal process is modeled by a mixed stochas…
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H40
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 91B70
Mathematics Subject Classification ID: 62F03
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 5218465
zbMATH Keywords: Fractional Brownian motion
zbMATH Keywords: Gaussian process
zbMATH Keywords: stochastic integration
zbMATH Keywords: Itô's formula
zbMATH Keywords: Girsanov's theorem
zbMATH Keywords: white noise
zbMATH Keywords: filtering problem
zbMATH Keywords: Black-Scholes model
zbMATH Keywords: hypotheses testing
author: Yuliya S. Mishura
reviewed by: Pavel Gapeev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-540-75873-0
OpenAlex ID: W588973965
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description: scientific article