Parameter estimation for Fisher–Snedecor diffusion

Meerschaert

Journal of Mathematical Analysis and Applications

2013

Pearson diffusions are governed by diffusion equations with polynomial coefficients. Fractional Pearson diffusions are governed by the corresponding time-fractional diffusion equation. They are useful for modeling sub-diffusive phenomena, caused by particle sticking and trapping. This paper provides explicit strong solutions for fractional Pearson diffusions, using spectral methods. It also presents stochastic solutions, using a non-Markovian inverse stable time change.

Section: Pearson Diffusions and Their Classificationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Fractional Pearson diffusions

Meerschaert

Journal of Mathematical Analysis and Applications

2013

“…The sixth subfamily (Student diffusion) also has a heavy-tailed stationary distribution, but its spectral properties differ from those of the FS and RG diffusions, and Student fractional diffusion will be dealt with in a separate paper. Results concerning the spectral representation of the transition density of Pearson diffusions and estimation of their parameters could be found in the series of papers (Avram et al (2011), 2012, 2013 a , b ), Leonenko & Šuvak (2010 a , b )). …”

Section: Pearson Diffusionsmentioning

confidence: 99%

Heavy-tailed fractional Pearson diffusions

Papić

Stochastic Processes and their Applications

et al. 2017

Self Cite

We define heavy-tailed fractional reciprocal gamma and Fisher-Snedecor diffusions by a non-Markovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher-Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal gamma and Fisher-Snedecor diffusions and strong solutions of the associated Cauchy problems for the fractional backward Kolmogorov equation.

“…If d 2 = 0, this is the Cox-Ingersoll-Ross (CIR) process, which is used in finance [15]. The study of Pearson diffusions began with Kolmogorov [20] and Wong [48], see also [1, 2, 17, 22, 23, 45]. Let p 1 ( x, t ; y ) denote the conditional probability density of x = X 1 ( t ) given y = X 1 (0), i.e., the transition density of this time-homogeneous Markov process.…”

Section: Introductionmentioning

confidence: 99%

Correlation structure of fractional Pearson diffusions

Meerschaert

Computers & Mathematics with Applications

2013

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 terms of Mittag-Leffler functions. That formula shows that fractional Pearson diffusions are long range dependent, with a correlation that falls off like a power law, whose exponent equals the order of the fractional derivative.