Density evolution in stochastic dynamical systems with memory: A universal algorithm

Stochastic dynamical systems with memory are usually modeled using stochastic functional differential equations. Quantifying the probability density evolution in these systems remains an open problem with strong practical applications. However, due to a lack of efficient methods for computing the probability density of stochastic functional differential equations in their general form, the application of these systems are severely restricted. We address this challenge by presenting a universal approach for computing the evolution of probability density in a broad class of stochastic dynamical systems with memory. The proposed approach approximates the stochastic functional equation via a discrete model derived from the Euler scheme and recursively estimates its probability density by computing that of the discretized counterpart. The method is deterministic and computationally efficient. To validate and demonstrate its effectiveness, we apply it to compute both transient and long-term probability density evolution for some typical climate models.