Heavy-tailed fractional Pearson diffusions.

Area of Science:

• Stochastic Processes
• Mathematical Physics
• Probability Theory

Background:

• Pearson diffusions are foundational models governed by backward Kolmogorov equations with polynomial coefficients.
• These diffusions find broad applications in various scientific and engineering fields.
• Existing models may not fully capture heavy-tailed phenomena crucial in many real-world systems.

Purpose of the Study:

• To define novel heavy-tailed fractional reciprocal gamma and Fisher-Snedecor diffusions.
• To investigate the properties of these new diffusion processes, particularly their heavy-tailed marginal distributions.
• To derive analytical solutions for the associated fractional backward Kolmogorov equations.

Main Methods:

• A non-Markovian time change was applied to standard Pearson diffusions.
• Fractional backward Kolmogorov equations were utilized to govern the new processes.
• Explicit expressions for transition densities were derived.
• Strong solutions for the Cauchy problems of the fractional backward Kolmogorov equation were obtained.

Main Results:

• The introduction of heavy-tailed fractional reciprocal gamma and Fisher-Snedecor diffusions.
• Demonstration that these diffusions possess heavy-tailed marginal distributions in the steady state.
• Derivation of explicit formulas for the transition densities of the newly defined diffusions.
• Establishment of strong solutions for the associated fractional backward Kolmogorov equations.

Conclusions:

• The study successfully defines and characterizes new heavy-tailed fractional diffusion processes.
• The derived explicit transition densities and solutions provide valuable tools for analyzing systems with heavy tails.
• These findings extend the theory of stochastic diffusions and offer new modeling capabilities.