Parameter estimation in fractional diffusion models
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 \(\theta a(s,X_s)\) is considered with known \(a(s,X_s)\) and unknown \(\theta\). Various estimators of \(\theta\) are proposed. In particular, for diffusion model driven by the Wiener process a general condition for the strong consistency of the maximum likelihood estimator of the drift is obtained. Chapter 6 is devoted to parameter estimation of some diffusion models driven by continuous time Gaussian processes which increments are not stationary. The trajectories of these processes are Hölder continuous up to the Orey index. The Orey index of fBm is equal to \(H\). Subfractional Brownian motion and bifractional Brownian motion are examples. Appendices A and B contain necessary selected facts from mathematical and functional analysis, probability, stochastic processes and stochastic calculus.
scientific article; zbMATH DE number 6811259