Parameter estimation in fractional diffusion models
type: Item
instance of: scholarly article
title: Parameter estimation in fractional diffusion models
zbMATH Open document ID: 1388.60006
DOI: 10.1007/978-3-319-71030-3
author: K. Kubilius
author: K. V. Ral'chenko
author: Yuliya S. Mishura
published in: Bocconi \& Springer Series
publication date: 2017-11-22T00:00:00Z
review text: The book under review is devoted to parameter estimation of the fractional diffusion models which are mostly described by the stochastic differential equation (SDE) \[ X_t=x_0+\int_0^t a(s,X_s) ds+ \int_0^t b(s,X_s) dB_s^H. \] Here, \(B_t^H\), \(t\geq 0\) is a fractional Brownian motion (fBm) which is a self-similar continuous time Gaussian process with stationary increments \(B^H_{t+s}-B^H_t\). Properties of \(B_t^H\) depend on the Hurst index \(H\) (\(0 <H <1\)). If \(H=1/2\) then \(B_t^H\) is a Wiener process \(W_t\) with uncorrelated increments. If \(H < 1/2\) or \(H > 1/2\) then increments of \(B_t^H\) are negatively or positively correlated, respectively. Diffusion models with correlated increments are widely used in natural sciences, financial and actuarial mathematics, computer networks, etc. Therefore, the book presents various diffusion models with memory and provides effective methods for parameter estimation in these models. It may be interesting for researchers and practitioners, graduate and postgraduate students. In Chapter 1, the basic properties of fBm are considered. General approach to diffusion models research is presented. Some classical SDEs are described in detail, e.g. Ornstein-Uhlenbeck equation. In Chapter 2, the Hurst index estimation of the fBm based on the asymptotic behavior of quadratic variations is presented. Chapter 3 is devoted to the Hurst index estimation from the solution of SDEs driven by fBm. Some strongly consistent and asymptotically normal estimators are proposed. In Chapter 4, mixed stochastic process \(M_t^H=a B_t^H+b W_t\) is investigated where \(B_t^H\) is fBm, \(W_t\) is a Wiener process and \(a\), \(b\) some coefficients. Strongly consistent estimators of parameters \(H\), \(a\) and \(b\) are proposed. Chapter 5 is devoted to the drift parameter estimation in the diffusion, fractional diffusion and mixed Brownian -- fractional Brownian diffusion models. In this chapter, instead of \(a(s,X_s)\) the drift \(\theta a(s,X…
reviewed by: Alex V. Kolnogorov
Mathematics Subject Classification ID: 60-02
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 62G05
zbMATH DE Number: 6811259
zbMATH Keywords: fractional diffusion models
zbMATH Keywords: stochastic differential equations
zbMATH Keywords: Hurst index estimation
zbMATH Keywords: drift parameter estimation
zbMATH Keywords: Orey index
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-319-71030-3
OpenAlex ID: W2782077593
description: scientific article; zbMATH DE number 6811259