Fractional Pearson Diffusions.

Correlation structure of fractional Pearson diffusions

The stochastic solution to a diffusion equations with polynomial coefficients is called a Pearson diffusion. If the first time derivative is replaced by a Caputo fractional derivative of order less than one, the stochastic solution is called a fractional Pearson diffusion. This paper develops an explicit formula for the covariance function of a fractional Pearson diffusion in steady state, in t...

The Pearson diffusions: A class of statistically tractable diffusion processes

The Pearson diffusions is a flexible class of diffusions defined by having linear drift and quadratic squared diffusion coefficient. It is demonstrated that for this class explicit statistical inference is feasible. Explicit optimal martingale estimating functions are found, and the corresponding estimators are shown to be consistent and asymptotically normal. The discussion covers GMM, quasi-l...

Positive time fractional derivative

In mathematical modeling of the non-squared frequency-dependent diffusions, also known as the anomalous diffusions, it is desirable to have a positive real Fourier transform for the time derivative of arbitrary fractional or odd integer order. The Fourier transform of the fractional time derivative in the Riemann-Liouville and Caputo senses, however, involves a complex power function of the fra...

Stochastic derivatives for fractional diffusions

In this paper, we introduce some fundamental notions related to the so-called stochastic derivatives with respect to a given σ-field Q. In our framework, we recall well-known results about Markov Wiener diffusions. We afterwards mainly focus on the case where X is a fractional diffusion and where Q is the past, the future or the present of X . We treat some crucial examples and our main result ...

Distributed-order fractional diffusions on bounded domains