Capturing non-Gaussian statistics in noise-injection driven dynamical systems
Zahra Vahdat, César Nieto, Abhyudai Singh · IFAC-PapersOnLine · 2024
The canonical approach to modeling stochasticity considers a dynamical system driven by Gaussian white noise. Here, we propose an alternative hybrid formulation, in which continuous dynamics is interspersed with noise injection events that occur at discrete times. The time interval between two successive events is drawn from an arbitrary, positively-valued continuous distribution. Although the hybrid approach converges to the classical stochastic differential equation when events occur sufficiently fast, the hybrid formulation captures a wider range of stochastic phenomena that deviate from Gaussian statistics when events occur at slower timescales. These deviations are mathematically captured by exact analytical expressions for higher-order moments, such as skewness and kurtosis of the state space. We illustrate this approach using the example of a nanosensor impacted by collisions from surrounding molecules. Our results provide an important generalization to capture the stochastic dynamics of physical, biological, and engineering systems and reveal a novel approach for exploiting higher-order moments to infer parameters that are not observable in lower-order moments.