Stability theory of stochastic differential systems

Stability in the Lyapunov sense for deterministic systems has been well studied since the beginning of this century. Since the middle of this century, Lyapunov method has been used to study stability and controllability of stochastic differential systems both in finite and infinite dimensional spaces. The sufficient and necessary conditions for exponential stability in [italic]p-th mean of It̂o type nonlinear stochastic differential equations was obtained. At the same time, practically stability of deterministic differential systems was also studied. Since the late 80's, practical stability in mean of stochastic differential systems has been studied. The theory of stochastic integral of processes which are not necessarily adapted has been developed by many authors. The stochastic Volterra equations have been discussed recently, which arise in applications particularly in finance theory. The present thesis studies the stability in several directions. Firstly, weak exponential stability of nonlinear It̂o type stochastic differential systems and related controlled problems are discussed, the conditions for this kind of stability are obtained. Secondly, exponential stability of Co-semigroup on Hilbert space is studied, a necessary and sufficient condition is given, which is used to prove the equivalency of weak exponential stability and exponential stability of some kind of linear stochastic evolution equations on a Hilbert space. Thirdly, practical stability and controllability of both It̂o type and L[acute]evy type stochastic systems are studied. Finally, conditional stability of stochastic Volterra equations is discussed. The concept of conditional stability, con-ditional uniformly asymptotically stability and weak conditional exponential stability are introduced. Some sufficient conditions of these types of stability are given