Stability Theory

The tightness assumptions (A4.1.6) or (A4.4.2) on {zϵ(t), t < ∞, ϵ > 0} or of (A4.4.1) on {xϵ(t),t < ∞, ϵ > 0} are essentially questions of stochastic stability. Of course, if the state spaces are bounded, then the cited as­sumptions are automatically satisfied. The deterministic specialization of the above cited tightness requirements is that the trajectories of interest (those of xϵ(ּ) and/or xϵ(ּ)) be bounded on the time interval of interest. To prove that boundedness in particular cases for the deterministic problem, some form of Liapunov function method is usually used. Stochastic “Liapunov function methods” are also very useful (if not indispensible at this time) to prove the required tightness for the stochastic problems, and we will discuss several approaches in this chapter.