The goal of this study is to introduce a unified computational framework for simulating random iteration equations (RIE), understood as iteration equations containing random variables. The novelty of this work is that full probability densities of the state vectors are propagated stepwise through the iterations avoiding the need of repetitive pathwise Monte Carlo simulations of the iteration equation. The presentation of the methodology is conceptually efficient based on recent work on static random equations and intentionally accessible. The technical requirements on the RIE are minimal based on the previous work, allowing for potential nonlinearities, discontinuities and stochasticities in the transfer function, as well as nonstandard densities and diffusion processes. As results, illustrative applications of random and stochastic differential equation simulations, a novel full-density gradient descent method (FDGD) for global optimization under uncertainty and examples of chaotic mappings are presented in order to demonstrate the breadth of the utility of this framework. In total, the character of the presentation is explorative and encourages new applications and theoretical studies.

翻译：本研究旨在提出一个统一的随机迭代方程（RIE）仿真计算框架，其中随机迭代方程指代包含随机变量的迭代方程。本工作的创新之处在于通过逐次迭代传播状态向量的全概率密度，避免了传统逐路径蒙特卡罗模拟迭代方程的冗余计算。基于近期关于静态随机方程的研究成果，本方法论在概念上高效且易于理解。基于前期工作，该方法对随机迭代方程的技术要求极低，允许传递函数存在非线性、间断性与随机性，同时支持非标准概率密度与扩散过程。作为应用示范，本文展示了随机与随机微分方程仿真、不确定性下全局优化的新型全概率密度梯度下降法（FDGD），以及混沌映射案例，以验证该框架的广泛适用性。总体而言，本研究的探索性特征将推动新应用与理论研究的开展。