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2017 | Buch

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Parameter Estimation in Fractional Diffusion Models

verfasst von: Prof. Dr. Kęstutis Kubilius, Prof. Dr. Yuliya Mishura, Prof. Kostiantyn Ralchenko

Verlag: Springer International Publishing

Buchreihe : Bocconi & Springer Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book is devoted to parameter estimation in diffusion models involving fractional Brownian motion and related processes. For many years now, standard Brownian motion has been (and still remains) a popular model of randomness used to investigate processes in the natural sciences, financial markets, and the economy. The substantial limitation in the use of stochastic diffusion models with Brownian motion is due to the fact that the motion has independent increments, and, therefore, the random noise it generates is “white,” i.e., uncorrelated. However, many processes in the natural sciences, computer networks and financial markets have long-term or short-term dependences, i.e., the correlations of random noise in these processes are non-zero, and slowly or rapidly decrease with time. In particular, models of financial markets demonstrate various kinds of memory and usually this memory is modeled by fractional Brownian diffusion. Therefore, the book constructs diffusion models with memory and provides simple and suitable parameter estimation methods in these models, making it a valuable resource for all researchers in this field.

The book is addressed to specialists and researchers in the theory and statistics of stochastic processes, practitioners who apply statistical methods of parameter estimation, graduate and post-graduate students who study mathematical modeling and statistics.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Description and Properties of the Basic Stochastic Models

Abstract

In this chapter we give definitions of the main stochastic processes and models used in the book and describe their properties. The statistical aspects of these models will be studied in the subsequent chapters.

Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

Chapter 2. The Hurst Index Estimators for a Fractional Brownian Motion

Abstract

We state again that the phenomenon of long range dependence is observed in various fields, such as hydrology, biology, finance, economy, chemistry, physics, telecommunication networks and others. Since a long memory parameter (or the Hurst index/exponent, usually denoted H) determines the mathematical properties of the model, its estimation is of great importance. One of the main methods of the estimation of the Hurst index involves the convergent sequence of the quadratic variations of the underlying process.

Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

Chapter 3. Estimation of the Hurst Index from the Solution of a Stochastic Differential Equation

Abstract

The present chapter is devoted to the Hurst parameter estimation constructed from the solutions to SDE driven by a fBm. Our goal is to construct strongly consistent and asymptotically normal estimators of the Hurst parameter H based on discrete observations of the underlying processes. In Sect. 3.1 we propose strongly consistent estimator of the Hurst index and specify the rate of convergence of this estimator to the true value of H when the diameter of partition of observation interval tends to zero. This estimator preserves its properties, if we replace the solution with its Euler-Peano approximation.

Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

Chapter 4. Parameter Estimation in the Mixed Models via Power Variations

Abstract

The purpose of this chapter is to develop parameter estimation for the models that, while being simple enough, already depart from the canons of self-similarity. They can take into account both the independence of process increments over short time intervals and the availability of memory at longer intervals. In fact, this is the simplest version of the multi-fractional Brownian motion.

Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

Chapter 5. Drift Parameter Estimation in Diffusion and Fractional Diffusion Models

The present chapter is devoted to the drift parameter estimation in the diffusion, fractional diffusion and mixed Brownian-fractional Brownian diffusion models. More precisely, we consider the solutions of SDEs involving Wiener process, fractional or mixed fractional Brownian motion with the linear unknown drift parameter to be estimated. In the case when fBm is involved, we always assume Hurst index to be known.

Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

Chapter 6. The Extended Orey Index for Gaussian Processes

Abstract

Stationarity of the increments of fBm is a useful feature in certain applications. However, there are cases when this property is undesirable. In order to enlarge the variety of models to choose from, extensions of fBm have been introduced recently by Houdré and Villa [70] (bifractional Brownian motion) and Bojdecki et al. [27] (sub-fractional Brownian motion). These processes share with fBm such properties as self-similarity, Gaussian property and others, however they do not have stationary increments and possess some new features. Immediately the question arises about the estimation of the parameters of such processes. On tools for statistical estimation, we hope that the reader learned from the Chaps. 2‐4 that one of the most important tools is quadratic variation. Except other applications, the asymptotic behavior of the quadratic variation leads to good results in estimation theory. As it was already mentioned in Chap. 2, the problem of the almost sure convergence of a quadratic variation has been solved for a wide class of processes by Baxter [10], Gladyshev [63], Klein and Giné [89], Bégyn [11], Malukas [114] etc.

Kęstutis Kubilius, Yuliya Mishura, Kostiantyn Ralchenko

Backmatter

Titel: Parameter Estimation in Fractional Diffusion Models
verfasst von: Prof. Dr. Kęstutis Kubilius
Prof. Dr. Yuliya Mishura
Prof. Kostiantyn Ralchenko
Copyright-Jahr: 2017
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-71030-3
Print ISBN: 978-3-319-71029-7
DOI: https://doi.org/10.1007/978-3-319-71030-3