Closed-form solutions for the Lévy-stable distribution

Area of Science:

• Mathematics and Physics
• Statistical Mechanics
• Financial Mathematics

Background:

• The Lévy-stable distribution models power-law phenomena with infinite variance, essential in economics and statistical mechanics.
• Existing numerical methods for this distribution face challenges due to its complex, non-explicit form.
• A uniform analytical solution has been lacking, hindering broader application.

Purpose of the Study:

• To develop a novel, uniform analytical approximation for the Lévy-stable distribution.
• To address the computational challenges associated with the non-explicit nature of this distribution.
• To provide a more accessible and accurate method for researchers utilizing Lévy-stable distributions.

Main Methods:

• The study introduces a 'trans-stable' auxiliary function to manage numerical calculation issues.
• A uniform solution is derived by asymptotically matching 'inner' and 'outer' power series expansions.
• The proposed analytical approximation is validated against numerical results.

Main Results:

• A new uniform analytical approximation for the Lévy-stable distribution has been successfully developed.
• The trans-stable function effectively resolves numerical instabilities in calculations.
• The approximation demonstrates high accuracy when compared to established numerical methods.

Conclusions:

• The presented uniform analytical approximation offers a robust and accurate method for the Lévy-stable distribution.
• This approach simplifies complex calculations, making the distribution more accessible for various applications.
• The findings are significant for fields relying on power-law modeling, including finance and anomalous transport.