Finite Time Stability of Discrete-Time Stochastic Dynamical Systems

In this paper, we address finite time stability in probability of discrete-time stochastic dynamical systems. Specifically, a stochastic comparison lemma is constructed along with a scalar system involving a generalized deadzone function to establish almost sure convergence and finite time stability in probability. This result is used to provide Lyapunov theorems for finite time stability in probability for Ito-type stationary nonlinear stochastic difference equations involving conditions on the minimum of the Lyapunov function itself along with a fractional power of the Lyapunov function. In addition, we establish sufficient conditions for almost sure lower semicontinuity of the stochastic settling-time capturing the average settling time behavior of the discrete-time nonlinear stochastic dynamical system.