Asymptotic Behavior Of Solutions To Stochastic Functional Differential Equations With Markovian Switching

The hybrid systems driven by continuous-time Markovian chains have been used tomodel many practical systems where they may experience abrupt changes in their structureand parameters caused by phenomena such as component failures or repairs, changingsubsystem interconnections, and abrupt environmental disturbances. The mathematicalmodels studied in this paper are stochastic functional differential equations withMarkovian switching, including stochastic differential delay equations with Markovianswitching and stochastic delay Kolmogorov systems with Markovian switching.The key issues of stochastic differential equations are the long-term behavior, thatis, the asymptotic properties of their solutions, including moment boundedness, momentboundedness average in time of the solutions. That is also the main study of the paper.The main achievements of the paper consist of three parts. First, a general conditionis given to ensure the existence, uniqueness, moment boundedness and moment boundednessaverage in time of the global solutions to stochastic functional differential equationswith Markovian switching. A group of conditions are imposed on the coefficients of theequations to ensure the general condition above holds.Second, a general sufficient condition is repectively given to ensure the existence,uniqueness and moment boundedness of the global positive solutions to stochastic functionaldifferential equations with Markovian switching. And a group of conditions areimposed on the coefficients of the equations to ensure the general sufficient conditionsabove holds.The two methods above are employed to stochastic differential delay equations withMarkovian switching.Third, a group of conditions are imposed on the coefficients of stochastic Kolmogorovsystems with Markovian switching to ensure the existence, uniqueness andmoment boundedness of the global positive solutions of the equations. These methodsare employed to stochastic delay Kolmogorov systems.In this paper, a general lemma is given to ensure the global solutions of stochasticfunctional differential equations with Markovian switching with properties of existence, uniqueness and moment boundedness. The existence and uniqueness of the solutionsare no longer an assumed condition, but one of the studies of the paper. The difficultiesbrought about by the addictive items of the generalized It(?) formula are solved byemploying the non-negative property of the density matrix with Markovian switching.