Quantifying Uncertainties in Complex Systems

TL;DR

This paper reviews recent methods for estimating parameters and quantifying uncertainties in complex stochastic systems modeled by differential equations driven by various types of noise.

Contribution

It provides a comprehensive overview of recent advances and algorithms for parameter estimation in stochastic differential equations with different noise processes.

Findings

Numerical algorithms are consistent with theoretical analysis.

Algorithms for estimating Hurst parameter and characteristic exponent are reviewed and implemented.

The paper consolidates methods for uncertainty quantification in complex stochastic models.

Abstract

Uncertainties are abundant in complex systems. Mathematical models for these systems thus contain random effects or noises. The models are often in the form of stochastic differential equations, with some parameters to be determined by observations. The stochastic differential equations may be driven by Brownian motion, fractional Brownian motion or L\'evy motion. After a brief overview of recent advances in estimating parameters in stochastic differential equations, various numerical algorithms for computing parameters are implemented. The numerical simulation results are shown to be consistent with theoretical analysis. Moreover, for fractional Brownian motion and $\alpha-$stable L\'evy motion, several algorithms are reviewed and implemented to numerically estimate the Hurst parameter $H$ and characteristic exponent $\alpha$.