Density Evolution for Stochastic Dynamical Systems Driven by α-Stable Processes: An Equation-Free Numerical Approach

While well-established approaches exist for computing probability densities of stochastic differential equations driven by Brownian motion, computing probability densities for general stochastic differential equations driven by α-stable processes remains challenging, and few general-purpose methods are available. A major difficulty is that the corresponding Fokker–Planck equations governing the density are generally unavailable in explicit form. In this paper, we develop an efficient approach to compute probability densities for stochastic differential equations driven by α-stable processes without relying on Fokker–Planck equations or any other governing equations. The proposed approach is easy to implement and computationally efficient. Numerical examples demonstrate its effectiveness and indicate an approximately first-order convergence rate with respect to the time step.