Statistical inference for fractional diffusion processes by B. L. S. Prakasa Rao

By B. L. S. Prakasa Rao

Stochastic strategies are frequent for version construction within the social, actual, engineering and existence sciences in addition to in monetary economics. In version development, statistical inference for stochastic techniques is of significant value from either a theoretical and an functions standpoint.

This e-book offers with Fractional Diffusion strategies and statistical inference for such stochastic procedures. the focus of the publication is to contemplate parametric and nonparametric inference difficulties for fractional diffusion techniques while a whole course of the method over a finite period is observable.

Key beneficial properties:

  • Introduces self-similar approaches, fractional Brownian movement and stochastic integration with appreciate to fractional Brownian movement.
  • Provides a finished evaluate of statistical inference for techniques pushed via fractional Brownian movement for modelling lengthy diversity dependence.
  • Presents a research of parametric and nonparametric inference difficulties for the fractional diffusion approach.
  • Discusses the fractional Brownian sheet and limitless dimensional fractional Brownian movement.
  • Includes contemporary effects and advancements within the region of statistical inference of fractional diffusion strategies.

Researchers and scholars engaged on the records of fractional diffusion procedures and utilized mathematicians and statisticians excited about stochastic procedure modelling will reap the benefits of this publication.

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Extra resources for Statistical inference for fractional diffusion processes

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The natural filtration of a process is understood as the P -completion of the filtration generated by this process. Let W H = {WtH , t ≥ 0} be standard fBm with Hurst parameter H ∈ (0, 1), that is, a Gaussian process with continuous sample paths such that W0H = 0, E(WtH ) = 0 and E(WsH WtH ) = 1 2H [s + t 2H − |s − t|2H ], t ≥ 0, s ≥ 0. 66) where C = {C(t), t ≥ 0} is an (Ft )-adapted process and B(t) is a non vanishing, non random function. 67) driven by fBm W H . 68) is not a stochastic integral in the Ito sense, but one can define the integral of a deterministic function with respect to fBm as the integrator in a natural sense FRACTIONAL BROWNIAN MOTION 25 (cf.

Define t Zt = 0 kH (t, s) dXs , t ≥ 0. 75) in Chapter 1. Let PθT be the measure induced by the process {Xt , 0 ≤ t ≤ T } when θ is the true parameter. 4 T QH,θ (s)dZs − 0 1 2 T 0 Q2H,θ (s)dwsH . 10) Maximum likelihood estimation We now consider the problem of estimation of the parameter θ based on the observation of the process X = {Xt , 0 ≤ t ≤ T } and study its asymptotic properties as T → ∞. Strong consistency Let LT (θ ) denote the Radon–Nikodym derivative dPθT /dP0T . The maximum likelihood estimator (MLE) θˆT is defined by the relation LT (θˆT ) = sup LT (θ ).

13 by choosing µ = H − 12 . 13. Let t Mt = 0 w(t, s)dZs , t ≥ 0. 57) In particular, the process {Mt , t ≥ 0} is a zero mean martingale. Proof: From the properties of Wiener integrals, it follows that the process {Mt , t ≥ 0} is a Gaussian process with mean zero. Suppose that s < t. ), I[0,t] s = w(s, u)du 0 = c1 B 3 3 − H, − H s 2−2H 2 2 = c22 s 2−2H . 58) Here . denotes the usual inner product in L2((0,∞)) . Note that the last term is independent of t which shows that the process M has uncorrelated increments.

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