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Fractional Brownian motion (fBm) has been widely used to model a number of phenomena in diverse fields from biology to finance. This huge range of potential applications makes fBm an interesting object of study. Several approaches have been used to develop the concept of stochastic calculus for fBm. The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches. Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices. This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance.



Fractional Brownian motion

1. Intrinsic properties of the fractional Brownian motion

The aim of this book is to provide a comprehensive overview and systematization of stochastic calculus with respect to fractional Brownian motion. However, for the reader's convenience, in this chapter we review the main properties that make fractional Brownian motion interesting for many applications in different fields.
The main references for this chapter are [76], [156], [177], [195], [209], [215]. For further details concerning the theory and the applications of long-range dependence from a more statistical point of view, we also refer to [81].

Stochastic calculus

2. Wiener and divergence-type integrals for fractional Brownian motion

We start our tour through the different definitions of stochastic integration for fBmof Hurst index H ∈(0,1) with the Wiener integralssince they deal with the simplest case of deterministic integrands. We show how they can be expressed in terms of an integral with respect to the standard Brownian motion, extend their definition also to the case of stochastic integrands, and then proceed to define the stochastic integral by using the divergence operator. In both cases we need to distinguish between H >1/2 and H <1/2.
The main references for this chapter are [6], [7], [8], [54], [68], [71], [72], [75], [76].

3. Fractional Wick Itô Skorohod (fWIS) integrals for fBm of Hurst index H >1/2

In this chapter we introduce the definition of stochastic integral with respect to the fBmfor Hurst index 1/2 <H <1 by using the white noise analysis method. At this purpose we define the fractional white noiseand stochastic integral as an element in the fractional Hida distribution space.
To obtain a classical Itô formula, we need the stochastic integral to be an ordinary random variable. Hence the ø-derivative is introduced to handle the existence of the Wick product in L2. Classical Itô type formulas are obtained and applications are discussed. The main references for this chapter are [32], [83] and [121].

4. WickItô Skorohod (WIS) integrals for fractional Brownian motion

In this chapter we again study the stochastic integral for the fBmfollowing the white noise approach. However, the integral is defined here as an element in the classicalHida distribution space by using the white noise theoryand Malliavin calculus for standard Brownian motionintroduced in Appendix A. The main advantage of this method with respect to the one presented in Chapter 3 is that it permits to define the stochastic integral for any H∈(0,1). In addition, it doesn't require the introduction of the fractional white noise theory since it uses the well-established one for the standard case.
On the other side, the approach of Chapter 3 can be seen as more intrinsic. For a further discussion of the relation among these two types of integrals we refer to Chapter 6. The main references for this chapter are [34] and [89].

5. Pathwise integrals for fractional Brownian motion

6. A useful summary

In Chapters 2 to 5 we have presented several ways of introducing a stochastic calculus with respect to the fBm. We have already underlined the relations among these different approaches, but in our opinion it is convenient to provide here a comprehensive summary, including a further investigation of their analogies and differences.
Moreover, in this chapter we present a general overview of the Itô formulas for the different definitions of stochastic integral for fBmtogether with an investigation of their relations.

Applications of stochastic calculus

7. Fractional Brownian motion in finance

Without Abstract

8. Stochastic partial differential equations driven by fractional Brownian fields

This chapter is devoted to the study of stochastic Poisson and stochastic heat equations driven by fractional white noise. The equations are solved both in the setting of white noise analysis and the setting of L2 space. The main references for this chapter are [126] and [184].

9. Stochastic optimal control and applications

Stochastic control has many important applications and is a crucial branch of mathematics. Some textbooks contain fundamental theory and examples of applications of stochastic control theory for systems driven by standard Brownian motion (see, for example, [96], [97], [182], [231]). In this chapter we shall deal with the stochastic control problem where the controlled system is driven by a fBm.
Even in the stochastic optimal control of systems driven by Brownian motion case or even for deterministic optimal control the explicit solution is difficult to obtain except for linear systems with quadratic control. There are several approaches to the solution of classical stochastic control problem. One is the Pontryagin maximum principle, another one is the Bellman dynamic programming principle. For linear quadratic control one can use the technique of completing squares. There are also some other methods for specific problems. For example, a famous problem in finance is the optimal consumption and portfolio studied by Merton (see [162]), and one of the main methods to solve this problem is the martingale method combined with Lagrangian multipliers. See [135] and the reference therein.
The dynamic programming method seems difficult to extend to fBmsince fBm– and solutions of stochastic differential equations driven by fBm– are not Markov processes. However, we shall extend the Pontryagin maximum principle to general stochastic optimal control problems for systems driven by fBms. To do this we need to consider backward stochastic differential equations driven by fBm.

10. Local time for fractional Brownian motion

In this chapter we present the main results concerning the local time of the fBmand provide its chaos expansion. In addition, we investigate the definition and the properties of the weighted and renormalized self-intersection local time for fBmand present a Meyer Tanaka formula valid for every H ∈(0,1).
The main references for this part are [28], [62], [87], [100], [110], [120], [122], [123] and [177].


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