From a Hierarchy of Stochastic Differential Equations to a Hierarchy of Generalized Beta Distributions
从随机微分方程层级到广义贝塔分布层级
We introduce a mean-reverting stochastic differential equation with a three-component stochastic term and show that it generates a hierarchy of steady-state (stationary) distributions. At the top level, the hierarchy is described by a modified-Beta distribution, while one- and two-parameter reductions produce compact-support, power-law-tailed, and exponential-type limiting families within a single stochastic framework. We then construct two generalized extensions of this…