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 mild 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：本研究旨在引入一个统一的计算框架，用于模拟随机迭代方程（RIE，即包含随机变量的迭代方程）。本工作的创新之处在于，状态向量的全概率密度通过迭代逐步传播，从而避免了重复使用路径方式蒙特卡罗模拟迭代方程的需求。该方法论的阐述基于近期关于静态随机方程的研究，在概念上高效且易于理解。根据先前工作，对RIE的技术要求较为宽松，允许传递函数存在潜在的非线性、不连续性和随机性，以及非标准密度和扩散过程。作为成果，本文展示了随机微分方程和随机微分方程模拟的应用实例、一种用于不确定性下全局优化的新型全密度梯度下降法（FDGD）以及混沌映射的例子，以证明该框架的广泛适用性。总体而言，本文的阐述具有探索性，并鼓励新的应用和理论研究。