  This paper studies stochastic boundedness of trajectories of a nonvanishing
stochastically perturbed stable LTI system. First, two definitions on
stochastic boundedness of stochastic processes are presented, then the
boundedness is analyzed via Lyapunov theory. In this proposed theorem, it is
shown that under a condition on the Lipchitz constant of the perturbation
kernel, the trajectories remain stochastically bounded in the sense of the
proposed definitions and the bounds are calculated. Also, the limiting behavior
of the trajectories have been studied. At the end an illustrative example is
presented, which shows the effectiveness of the proposed theory.
