General decay stability of stochastic functional differential equations

In this thesis, pth moment and almost sure stability on a general decay rate for several types of stochastic functional differential equations were studied. By applying Razumikhin method and Lyapunov method stability criteria were obtained. Having in mind that some types of stochastic differential equations are not exponentially stable, the information about the stability with respect to a certain lower decay rate is very important, which was the motive for research in this paper. Some future research could focus on application of Krasovskii-Lyapunov method for exploring the general decay stability of the already studied types of stochastic differential equations. In this way, we could get different stability and decay rate criteria with respect to those obtained in the thesis. The research based on the modified results of this thesis could be continued for studying stability of various classes of stochastic functional differential equations with respect to martingale and martingale measures. The research on pth moment and almost sure stability and pth moment instability on a general decay rate for stochastic functional differential equations with Markovian switching and delayed impulses could be extended to stochastic differential equations with random impulses and Markovian switching which more realistically describe processes impulsive in kind. Also, Razumikhin method and Lyapunov method could be applied in studying pth moment and almost sure stability on a general decay rate for hybrid impulsive stochastic differential equations with switching not defined by Markov chain law as well as stochastic neural networks. Moreover, these methods could be used for studying general decay stability of all the above mentioned types of impulsive stochastic differential equations with respect to martingale and martingale measures. Some future research could be based on the application of LMI theory results to studying pth moment and almost sure stability on a general decay rate of perturbed impulsive stochastic functional differential equations with Markovian switching and hybrid perturbed impulsive stochastic functional differential equations