Non-stationary Feller process with time-varying coefficients

TL;DR

This paper analyzes a non-stationary Feller process with time-varying coefficients, deriving its exact distribution, asymptotic behavior, and potential applications, highlighting its dynamic accessibility to the origin.

Contribution

It provides the first exact distribution, characteristic function, and cumulants for the non-stationary Feller process with time-dependent parameters.

Findings

Exact probability distribution and characteristic function derived.

Density approaches a time-varying Gamma distribution over long times.

Process exhibits a dynamic power-law behavior near the origin.

Abstract

We study the non-stationary Feller process with time varying coefficients. We obtain the exact probability distribution exemplified by its characteristic function and cumulants. In some particular cases we exactly invert the distribution and achieve the probability density function. We show that for sufficiently long times this density approaches a Gamma distribution with time-varying shape and scale parameters. Not far from the origin the process obeys a power law with an exponent dependent of time, thereby concluding that accessibility to the origin is not static but dynamic. We finally discuss some possible applications of the process.