Nonstationary Feller process with time-varying coefficients

Area of Science:

• Stochastic Processes
• Probability Theory
• Mathematical Physics

Background:

• Feller processes are fundamental in modeling random phenomena.
• Nonstationary processes with time-varying coefficients present unique analytical challenges.
• Understanding their behavior is crucial for applications in finance, physics, and biology.

Purpose of the Study:

• To investigate the properties of the nonstationary Feller process with time-varying coefficients.
• To derive the exact probability distribution and density function.
• To analyze the process's behavior over time, including its approach to the origin and its long-term distribution.

Main Methods:

• Derivation of the characteristic function and cumulants for the exact probability distribution.
• Exact inversion of the distribution in specific cases to obtain the probability density function.
• Asymptotic analysis to determine the long-time behavior and near-origin dynamics.

Main Results:

• The exact probability distribution was obtained via characteristic function and cumulants.
• The probability density function was derived for particular cases.
• For long times, the process approaches a Gamma distribution with time-varying parameters.
• Near the origin, the process exhibits a time-dependent power-law behavior, indicating dynamic accessibility.

Conclusions:

• The nonstationary Feller process exhibits complex dynamics, including time-varying convergence to a Gamma distribution.
• Accessibility to the origin is not static but evolves with time.
• The study provides a theoretical framework for understanding and applying such processes in various scientific fields.