A Full-Density Approach to Simulating Random Iteration Equations with Applications

TL;DR

This paper introduces a unified computational framework for simulating random iteration equations by propagating full probability densities, enabling efficient analysis of stochastic systems without repetitive Monte Carlo simulations.

Contribution

It presents a novel full-density propagation method for RIEs that handles nonsmooth nonlinearities and stochasticities, expanding simulation capabilities beyond traditional approaches.

Findings

Demonstrated applications in nonlinear stochastic differential equations

Developed a full-density gradient descent method for optimization under uncertainty

Showcased examples of chaotic mappings and stochastic processes

Abstract

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. Based on previous work, the modeling requirements for RIEs allow for potential nonsmooth nonlinearities and stochasticities in the transfer function, as well as nonstandard probability densities and diffusion processes. As results, illustrative applications of nonlinear 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.