Fractional Brownian motion (fBm) B^H = [, t ≥ 0} with Hurst index H ∈ (0, 1) is a very interesting stochastic object that has attracted increased attention due to its peculiar properties. On the one hand, this is a Gaussian random process with a fairly simple covariance function that provides the Hölder property of trajectories up to the order H. On the other hand, it is a generalization of the Wiener process, which corresponds to the value of the Hurst index H = 1/2. Finally, it is neither a process with independent increments, nor a Markov process, nor a semimartingale unless H = 1/2, and therefore it can be used to model quite complex real processes that demonstrate the phenomenon of memory, both long and short. Long memory corresponds to H > 1/2, while short memory is inherent in H < 1/2. The combination of these properties is useful in modeling the processes occurring in devices that provide cellular and other types of communication, in physical and biological systems and in finance and insurance. Thus, the fBm itself deserves special attention. We will not discuss all the aspects of fBm here, and recommend the books [BIA 08, KUB 17, MIS 08, MIS 17, MIS 18, NOU 12, NUA 03, SAM 06] for more detail concerning various fractional processes.

Now note that the absence of semimartingale and Markov properties always causes the study of the possibility of the approximation of fBm by simpler processes, in a suitable metric. Without claiming a comprehensive review of the available results, we list the following studies: approximation of fBm by the continuous processes of bounded variation was studied in [AND 06, RAL 11b], approximating wavelets were considered in [AYA 03], weak convergence to fBm in the schemes of series of various sequences of processes was discussed in [GOR 78, NIE 04, TAQ 75] and some other studies, and summarized in [MIS 08]. The paper [MUR 11] contains a presentation of fBm in terms of an infinite-dimensional Ornstein-Uhlenbeck process. The approximation of fBm by semimartingales is proposed in [DUN 11]. The article [RAL 11a] investigates smooth approximations for the so-called multifractional Brownian motion, a generalization of fBm to the case of time-varying Hurst index. Approximation of fBm using the Karhunen theorem and using various decompositions into series over functional bases is also investigated in great detail.

There is also such a question, which, in fact, served as the main incentive for writing this book: is it possible to approximate an fBm by martingales, in a reasonably chosen metric? If not, is it possible to find a projection of fBm on the class of martingales and the distance between fBm and this projection? Such a seemingly simple and easily formulated question actually led to, in our opinion, quite unexpected, non-standard and interesting results that we decided to offer them to the attention of the reader. Metric, which was proposed, has the following form:

where M = [M_t, t ∈ [0, T]} is a martingale adapted to the filtration generated by B^H. So, we consider the distance in the space L_∞([0,T]; L₂(Ω)). The first problem, considered in this book, is the minimization of ρH(M) over the class of adapted martingales. Chapter 1 is fully devoted to this problem. We perform the following procedures step by step: introducing the so-called Molchan representation of fBm via Volterra kernel and the underlying Wiener process; proving that minimum is achieved within the class of martingales of the form , where W is the underlying Wiener process and a is a non-random function from L₂([0,T]). As a result, the minimization problem becomes analytical. Since it is essentially minimax problem, we used a convex analysis to establish the existence and uniqueness of minimizing function a. The existence follows from the convexity of the distance. However, the proof of the uniqueness essentially relies on self-similarity of fBm. If some other Gaussian process is considered instead of fBm, the minimum of the distance may be attained for multiple functions a. Then, we propose an original probabilistic representation of the minimizing function a and establish several properties of this function. However, its analytical representation is unknown; therefore, the problem is to find its values numerically. In this connection, we considered a discrete-time counterpart of the minimization problem and reduced it, via iterative minimization using alternating minimization method, to the calculation of the Chebyshev center. It allows us to draw the plots of the minimizing function, as well as the plot of square distance between fBm and the space of adapted Gaussian martingales as a function of the Hurst index.

So, since the problem of finding a minimizing function in the whole class L₂([0,T]) turned out to be one that requires a numerical solution, and it is necessary to use fairly advanced methods, we then tried to minimize the distance of an fBm to the subclasses of martingales corresponding to simpler functions, in order to obtain an analytical solution, or numerical, but with simpler methods, without using tools of convex analysis. Since the Volterra kernel in the Molchan representation of fBm consists of power functions, it is natural to consider various subclasses of L₂([0,T]) consisting of power functions and their combinations. Even in this case, the problem of minimization is not easy and allows an explicit solution only in some cases, many of which are discussed in detail in Chapter 2. Somewhat unexpected, however, for some reason, natural, is the fact that the normalizing constant in the Volterra kernel, which usually does not play any role and is even often omitted, comes to the fore in calculations and, so to speak, directs the result. Moreover, in the course of calculations, interesting new relations were obtained for gamma functions and their combinations, and even a new upper bound for the cardinal sine function was produced.

Chapter 3 is devoted to the approximations of fBm by various processes of comparatively simple structure. In particular, we represent fBm as a uniformly convergent series of Lebesgue integrals, describe the semimartingale approximation of fBm and propose a construction of absolutely continuous processes that converge to fBm in certain Besov-type spaces. Special attention is given to the approximation of pathwise stochastic integrals with respect to fBm. In the last section of this chapter, we study smooth approximations of multifractional Brownian motion.

Appendix 1 contains the necessary auxiliary facts from mathematical, functional and stochastic analyses, especially from the theory of gamma functions, elements of convex analysis, the Garsia-Rodemich-Rumsey inequality, basics of martingales and semimartingales and introduction to stochastic integration with respect to an fBm. Appendix 2 explains how to evaluate the Chebyshev center, together with pseudocode. In Appendix 3, we describe several techniques of fBm simulation. In particular, we consider in detail the Cholesky decomposition of the covariance matrix, the Hosking method (also known as the Durbin-Levinson algorithm) and the very efficient method of exact simulation via circulant embedding and fast Fourier transform. A more detailed description of the book’s content by section is at the beginning of each chapter.

The results presented in this book are based on the authors’ papers [BAN 08, BAN 11, BAN 15, DOR 13, MIS 09, RAL 10, RAL 11a, RAL 11b, RAL 12, SHK 14] as well as on the results from [DAV 87, DIE 02, DUN 11, HOS 84, SHE 15, WOO 94].

It is assumed that the reader is familiar with the basic concepts of mathematical analysis and the theory of random processes, but we tried to make the book self-contained, and therefore most of the necessary information is included in the text. This book will be of interest to a wide audience of readers; it is comprehensible to graduate students and even senior students, useful to specialists in both stochastics and convex analysis, and to everyone interested in fractional processes and their applications.

We are grateful to everyone who contributed to the creation of this book, especially to Georgiy Shevchenko, who is the author of the results concerning the probabilistic representation of the minimizing function.

Oksana BANNA, Yuliya MISHURA, Kostiantyn RALCHENKO, Sergiy SHKLYAR

January 2019