Neutral Stochastic Functional Differential Equations

This paper investigate neutral stochastic functional differential equations, neutral stochastic functional differential equations with Poisson jumping and neutral stochastic functional differential equations with Markovian switching. Under non-Lipschitz condi-tions, the existence-and-uniqueness theorems of solutions for such neutral equations are established, where new approximations are used in the proof of theorems. In addition, the estimation of the solutions are presented. Moreover, using the Lyapunove function and gen-eralized Ito formula, the p moment stable, the almost sure stable and the boundedness with the general sense stability with the reference functionψγare discussed.The first chapter introduces the current situation and the development of stochastic dif-ferential equations, neutral stochastic functional differential equations and stochastic differ-ential equations with Markovian switching. Some applications problem with those equations are presented.Chapter 2 proves the existence-and-uniqueness theorems of neutral stochastic func-tional differential equations with non-Lipschitz condition and weaken linear growth condi-tion. Under more general non-linear growth condition, the solutions of such equations are discussed. Moreover, Razumikhin-type theorems onψγstability are established, the special cases are exponential stability and polynomial stability.Chapter 3 investigates neutral stochastic functional differential equations with Poisson jumping. Under non-Lipschitz condition and non-linear growth condition, this chapter es-tablishes the existence-and-uniqueness theorems. Moreover, the stability of the solutions depending the initial data is discussed. The corresponding example are presented to illus-trate the applications.Chapter 4 discusses neutral stochastic delay differential equations with Markovian switching. Under non-Lipschitz condition and linear growth condition, the existence-and-uniqueness theorems are proved. Moreover, applying generalize Ito formula, sufficient con-ditions are obtained for theψγbound and theψγstable. Some example are given to illustrate the results.Chapter 5 further studies infinite delay neutral stochastic functional differential equations with Markovian switching. Under non-Lipschitz condition and non-linear growth condition, the existence-and-uniqueness theorems are obtained. Razumikhin-type theorems on p-th momentψγstability are established. Using Borel-Cantelli lemma, sufficient conditions are obtained onψγtrajectory stability.