Stochastic Modeling and Analysis of Random Differential Equations

Stochastic models based on random differential equations introduce uncertainty directly into the formulation of dynamical systems by treating parameters, initial conditions or external forcings as random variables or processes. Solutions are no longer deterministic trajectories but stochastic processes described by their moments and full probability distributions. Key analytical and numerical approaches include Monte Carlo simulation, moment-closure methods, stochastic Galerkin projection with polynomial chaos expansions, Karhunen–Loève representations and random variable transformation techniques. These methods address challenges posed by high dimensionality, nonlinearity and non-Gaussian inputs. Random differential equations find applications across disciplines from climate extremes and structural dynamics to epidemiology, population biology and financial risk assessment, providing quantitative insights into variability, reliability and rare-event probabilities.

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Stochastic Modeling and Analysis of Random Differential Equations publication trend

The graph below shows the total number of articles in stochastic modeling and analysis of random differential equations across all publications each year (not limited to Nature Index journals).

Technical terms

Random Differential Equation (RDE): A differential equation in which parameters, initial conditions or forcing terms are modelled as random variables or stochastic processes, yielding a solution that is itself a stochastic process.

Random Variable Transformation (RVT) Technique: A method to derive the probability density function of a function of random inputs by applying change-of-variable formulas to their joint distribution.

Karhunen–Loève Expansion: A representation of a stochastic process as an infinite series of orthogonal deterministic functions weighted by uncorrelated random coefficients, facilitating dimensional reduction and efficient simulation.

Mean Square Convergence: A mode of convergence for stochastic processes in which the expected squared difference between an approximate solution and the true process tends to zero as the approximation is refined.

References

1. A note on the application of the RVT method to general classes of single-species population models formulated by random differential equations. Computational and Applied Mathematics (2024).
2. Modeling the biological growth with a random logistic differential equation. Environmental and Ecological Statistics (2023).
3. On the random wave equation within the mean square context. Discrete and Continuous Dynamical Systems - S (2022).