The goal of this study is to introduce a unified framework for simulating random iteration equations (RIE), understood as iteration equations containing random variables. The main idea is to propagate approximations of the full state density from one iteration to the next, rather than estimating it from many repeated pathwise Monte Carlo simulations. The presentation of the RIE modeling framework is conceptually simple based on recent work on static random equations and designed to be accessible. The modeling requirements for RIEs allow for potential nonsmooth nonlinearities and stochasticities in the transfer function. Additionally, the RIE computational strategy for full-density propagation is presented based on iterative likelihood / posterior calculations. As results, illustrative applications of nonlinear random and stochastic differential equation simulations, a new 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.

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