Sevearal Types Of Stochastic Differential Equations With Polynomial Growth

In the last years, several authors have studied that the environmental noise cansuppress explosions in population dynamics system. But there is little work on whetherenvironmental noise suppress the exponential growth of the solution of stochasticdifferential delay equation. In the paper, we will show that a delay differential systemwhose solution grows exponentially become a new system whose solution will grow atmost polynomially. In other words, we reveal that environmental noise will suppress theexponential growth.Besides, now most of the papers consider only the white noise but not colour noiseyet. In this paper we will consider a given system under regime switching and theenvironmental noise work together to make the system change significantly. In otherwords, we will consider the stability of the solution of stochastic differential delayequation with Markovian switching.Recently, neutral stochastic differential systems have been studied intensively. Theunderlying results only limit to discuss the moment boundedness, moment exponentialstability of neutral stochastic differential system. We may therefore wonder some ofpolynomially growing. In the paper, we will give a positive answer.To demonstrate the conclusion in the previous chapter, an example is written in thelast part of the paper .