Density Evolution for Stochastic Dynamical Systems Driven by α-Stable Processes: An Equation-Free Numerical Approach
Abstract
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.
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BibTeX
@misc{liu2026density,
title = {{Density Evolution for Stochastic Dynamical Systems Driven by α-Stable Processes: An Equation-Free Numerical Approach}},
author = {Yan Liu and Xu Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28243},
eprint = {2610.28243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28243},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Density Evolution for Stochastic Dynamical Systems Driven by α-Stable Processes: An Equation-Free Numerical Approach
AU - Yan Liu
AU - Xu Sun
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28243
UR - https://arxiv.org/abs/2610.28243
AB - 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.
ER -