Stationary distributions of a class of stochastic delay differential equations with Markovian switching
In stochastic analysis problems,many algorithms were based on the existence and uniqueness of the stationary distribution of stochastic systems.In this paper,the existence and uniqueness of stationary distributions for a class of stochastic delay differential equations with Markovian switching in Hilbert space was studied.The fundamental solutions of the corresponding deterministic equations of the studied equations were considered,and the solutions were expressed by using the constant variability formula.The assumptions required were given by using the weak convergence method and the properties of Markov chains,and the conditions for the existence of stationary distributions under Markov switching were established.Based on the assumptions,by means of Ito isometry formula,Gronwall lemma and exponential stability of the fundamental solution,the sufficient conditions for the existence and uniqueness of the stationary distribution of the stochastic delay differential equation were given and strictly proved.Conclusion improved and generalized the related results in the literature.
Markov switchstochastic delay differential equationstationary distributionfundamental solutionGronwall lemmaIto formula
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