Statistical solution to SDEs with α -stable Lévy noise via deep neural network

Hao Zhang, Yong Xu, Yongge Li, Jürgen Kurths

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

Abstract

The probability density function of the stochastic differential equations with α-stable Lévy noise is determined by the deterministic fractional Fokker–Planck (FFP) equation with the Riesz fractional derivative. In this paper, to solve the FFP equation, a new algorithm which is an efficient numerical approach using the deep neural network is proposed in view of the mesh-dependence phenomenon in the traditional discrete scheme and the nonsmooth solution of the Monte Carlo method. Under the framework of the DL-FP algorithm, the “fractional centered derivative” approach is applied to approximate the Riesz fractional derivative of the output in the neural network, which is the major novelty of this approach. Numerical results are presented to demonstrate the accuracy of the approach. Comparing with the traditional discrete scheme and the Monte Carlo method, the proposed algorithm is mesh-less and smoother. Furthermore, the proposed technique performs well in dealing with the heavy-tail case through comparing with the analytical solution.

Original languageEnglish
Pages (from-to)1129-1140
Number of pages12
JournalInternational Journal of Dynamics and Control
Volume8
Issue number4
DOIs
StatePublished - 1 Dec 2020

Keywords

  • Deep neural network
  • Fractional Fokker–Planck
  • Fractional centered derivative
  • Heavy-tail
  • Riesz fractional derivative

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