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

TL;DR

The paper addresses the generative mechanism of the $p$-order cloud model and shows how a reparameterization yields a stochastic recurrence equation with an absolute-value nonlinearity. It proves the existence and uniqueness of a stationary solution under standard conditions, and explicitly demonstrates that for a standard Gaussian driver $A$, the logarithmic moment satisfies $\mathbb{E}[\log|A|] = -\frac{\gamma + \log 2}{2} < 0$, ensuring almost sure convergence. This establishes a rigorous stochastic stability foundation for cloud models and supports uncertainty quantification in applications such as image processing, evaluation, and decision-making systems.

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.

Theorems & Definitions (9)

• Definition 1
• Definition 2
• Theorem 1: brandt1986stochastic
• Theorem 2
• proof
• Lemma 1
• proof
• Theorem 3: Existence and Uniqueness of Stationary Solution
• proof