Governing equations for Probability densities of stochastic differential equations with discrete time delays

Yayun Zheng; Xu Sun

doi:10.3934/dcdsb.2017182

Article Contents

2017, Volume 22, Issue 9: 3615-3628. Doi: 10.3934/dcdsb.2017182

This issue Previous Article Generalized Lyapunov-Razumikhin method for retarded differential inclusions: Applications to discontinuous neural networks Next Article

Governing equations for Probability densities of stochastic differential equations with discrete time delays

Yayun Zheng and
Xu Sun^,

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China

* Corresponding author: Xu Sun
* Corresponding author: Xu Sun

Received: August 31, 2016

Revised: March 31, 2017

Published: October 2017

The authors are supported by National Natural Science Foundation of China grants 11531006

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Abstract

The time evolution of probability densities for solutions to stochastic differential equations (SDEs) without delay is usually described by Fokker-Planck equations, which require the adjoint of the infinitesimal generator for the solutions. However, Fokker-Planck equations do not exist for stochastic delay differential equations (SDDEs) since the solutions to SDDEs are not Markov processes and have no corresponding infinitesimal generators. In this paper, we address the open question of finding governing equations for probability densities of SDDEs with discrete time delays. In the governing equation, densities for SDDEs with discrete time delays are expressed in terms of those for SDEs without delay. The latter have been well studied and can be obtained by solving the corresponding Fokker-Planck equations. The governing equation is given in a simple form that facilitates theoretical analysis and numerical computation. Some example are presented to illustrate the proposed governing equations.

Keywords:

Stochastic differential equations,
Brownian motions,
probability density,
discrete time delay,
stochastic delay differential equations.

Mathematics Subject Classification: Primary:60H10, 65C50;Secondary:34K50.

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