Moment stability of stochastic processes with applications to control systems

Abstract

We establish new conditions for obtaining uniform bounds on the moments of discrete-time stochastic processes. Our results require a weak negative drift criterion along with a state-dependent restriction on the centered conditional moments of the process. They, in particular, generalize the main result of [22] which requires a constant bound on the averaged one-step jumps of the process. The state-dependent feature of our results make them suitable for a large class of multiplicative-noise processes. Under the additional assumption of the Markovian property, we prove new results on ergodicity that do not rely on a minorization condition typically needed in ergodic theorems. Several applications to iterative systems, control systems, and other dynamical systems with state-dependent multiplicative noise are included, and these illustrative examples demonstrate the wide applicability of our results.