  This paper addresses stochastic stabilization in case where implementation of
control policies is digital, i. e., when the dynamical system is treated
continuous, whereas the control actions are held constant in predefined time
steps. In such a setup, special attention should be paid to the
sample-to-sample behavior of the involved Lyapunov function. This paper extends
on the stochastic stability results specifically to address for the
sample-and-hold mode. We show that if a Markov policy stabilizes the system in
a suitable sense, then it also practically stabilizes it in the sample-and-hold
sense. This establishes a bridge from an idealized continuous application of
the policy to its digital implementation. The central result applies to
dynamical systems described by stochastic differential equations driven by the
standard Brownian motion. Generalizations are discussed, including the case of
non-smooth Lyapunov functions for systems driven by bounded noise. A brief
overview of bounded noise models is given.
