Approximate solutions of stochastic differential delay equations with Markovian switching

Abstract

This version is available at https://strathprints.strath.ac.uk/16869/ Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaking activities or any commercial gain. You may freely distribute both the url (https://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to the Abstract. Our main aim is to develop the existence theory for the solutions to stochastic differential delay equations with Markovian switching (SDDEwMSs) and to establish the convergence theory for the Euler-Maruyama approximate solutions under the local Lipschitz condition. As an application, our results are used to discuss a stochastic delay population system with Markovian switching.

Approximate solutions for a class of doubly perturbed stochastic differential equations

“…If α = β = 0, then DPSDEwJs and DPSDEwMS will become SDEs with jumps and SDEs with Markovian switching which were investigated by [23][24][25][26][27][28][29][30][31][32][33][34]. Similarly, we can also give the Carathéodory approximate solution and show that the Carathéodory approximate solution converges to the solution of SDEs with jumps and SDEs with Markovian switching under our assumptions.…”

Permanence and asymptotical behavior of stochastic prey–predator system with Markovian switching

“…The existing general existence-and-uniqueness theorem on SDEwMSs (see e.g. [38]) is not applicable to this special SDEwMS, however, Theorem 3.2 in the literature [29] facilitates this work. A direct result from it is given in the following.…”

“…These conditions are somehow still too restrictive, for example, they are not satisfied by SDEwMS (1.7). Fortunately, literature [29] showed the convergence in probability of the EM approximate solution to the exact solution.…”

“…[15,25,36]). However there is few work on numerical method of SDEwMSs until the publications [29,38,39,51]. Most existing strong convergence theory for numerical methods requires f and g to be globally Lipschitz (see [15,25,36,38]).…”

“…Now we recall here the EM approximate solution definition given in [29,38,39,51]. Given a step size > 0, let X(t), Y(t) be the continuous EM approximate solution to the SDDEwMS (1.7).…”