# Statistical Inference for Fractional Diffusion Processes by B. L. S. Prakasa Rao

By B. L. S. Prakasa Rao

Stochastic procedures are universal for version development within the social, actual, engineering and existence sciences in addition to in monetary economics. In version construction, statistical inference for stochastic tactics is of serious significance from either a theoretical and an purposes perspective.

This publication offers with Fractional Diffusion approaches and statistical inference for such stochastic approaches. the focus of the ebook is to think about parametric and nonparametric inference difficulties for fractional diffusion techniques while an entire course of the method over a finite period is observable.

Key features:

• Introduces self-similar approaches, fractional Brownian movement and stochastic integration with appreciate to fractional Brownian motion.
• Provides a finished evaluate of statistical inference for techniques pushed by means of fractional Brownian movement for modelling lengthy diversity dependence.
• Presents a research of parametric and nonparametric inference difficulties for the fractional diffusion process.
• Discusses the fractional Brownian sheet and countless dimensional fractional Brownian motion.
• Includes fresh effects and advancements within the sector of statistical inference of fractional diffusion processes.

Researchers and scholars engaged on the information of fractional diffusion approaches and utilized mathematicians and statisticians all for stochastic approach modelling will reap the benefits of this book.

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Additional info for Statistical Inference for Fractional Diffusion Processes (Wiley Series in Probability and Statistics)

Example text

1 Introduction Statistical inference for diffusion-type processes satisfying SDEs driven by Wiener processes was studied earlier and a comprehensive survey of various methods is given in Prakasa Rao (1999a). There has been some recent interest in studying similar problems for stochastic processes driven by fBm. 2 SDEs and local asymptotic normality One of the basic tools in the study of asymptotic theory of statistical inference is the concept of local asymptotic normality. Several important properties of estimators of parameters involved in such processes follow as a consequence of the local asymptotic normality of the family of probability measures generated by the processes.

116) or equivalently t X(t) = X(0) + a t X(s)ds + b 0 0 X(s)dWsH , t ≥ 0. 117) It can be shown that the solution of the above SDE is X(t) = X(0) exp[at + bW H (t)], t ≥ 0. 118) Change-of-variable formula It is known that the chain rule dF (f (x)) = F (f (x))df (x) does not hold for functions f of Holder exponent 12 arising as sample paths of stochastic processes which are semimartingales. However, for functions of Holder exponent greater than 12 , the classical formula remains valid in the sense of Riemann–Stieltjes integration.

The techniques used are composition formulas and integration by parts rules for fractional integrals and fractional derivatives (cf. Samko et al . (1993)). Note that if f or g is a smooth function on a ﬁnite interval (a, b), the Lebesgue–Stieltjes integral can be written in the form b b f (x) dg(x) = a f (x)g (x)dx a or b a b f (x) dg(x) = − f (x)g(x)dx + f (b−)g(b−) − f (a+)g(a+). a Here f (a+) = limδ 0 f (a + δ) and g(b−) = limδ 0 f (b−δ) whenever the limits exist. The main idea of Zahle’s approach is to replace the ordinary derivatives by the fractional derivatives.