Stochastic calculus for fractional Brownian motion and related processes / Yuliya S. Mishura

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 stochastic d ifferential equation and the observation process is a sum of the fractional Brownian integral and the term of bounded variation, and the case of conditionally Gaussian linear systems with mixed signals and fractional Brownian observation. In both cases only non-random integrands are considered.

Chapter 5 studies the models of financial markets involving fractional Brownian motion. It develops some methods of construction of the long-memory arbitrage-free models and the discussion of different approaches to this problem. It contains extensions of the Black-Scholes model driven by mixed Brownian-fractional Brownian motion based on pathwise and Wick integration.

Chapter 6 solves some statistical problems in models involving fractional Brownian motion. Namely, it contains a solution of the problem of testing the complex hypotheses concerning the drift of the geometric Brownian motion, one of which corresponds to the pathwise integral and another one to the Wick integral. The properties of estimates of the drift parameter like local asymptotic normality and asymptotic efficiency in different models involving dependent fractional and standard Brownian motions are also studied. (Zentralblatt)