Stationary Solution of p-Order Cloud Model via Stochastic Recurrence Equation

TL;DR

This paper establishes the existence and uniqueness of stationary solutions for the p-order cloud model by reformulating it as a stochastic recurrence equation and analyzing its stability under certain conditions.

Contribution

It introduces a novel reformulation of the p-order cloud model as an SRE and proves conditions for its stochastic stability and stationarity.

Findings

Existence and uniqueness of stationary solutions are proven.

Logarithmic moment of the model's coefficient is negative, ensuring convergence.

Provides a rigorous foundation for uncertainty quantification in cloud models.

Abstract

This paper investigates the generative mechanism of the p-order cloud model, which is a mathematical framework for representing uncertainty with applications in image processing, evaluation, and decision-making systems. By employing a reparameterization technique, we reformulate the cloud model as a stochastic recurrence equation (SRE) with a nonlinear transformation involving an absolute value. Under standard assumptions of stationarity, ergodicity, and an appropriate integrability condition, we establish the existence and uniqueness of a stationary solution. In particular, we demonstrate that the logarithmic moment of the model's coefficient, modeled as a standard normal random variable, is negative, thereby ensuring almost sure convergence. These results provide new insights into the stochastic stability of cloud models and offer a rigorous foundation for further theoretical and practical developments in uncertainty quantification.