Some remarks on stochastic converse Lyapunov theorems

In this brief note, we investigate some constructions of Lyapunov functions for stochastic discrete-time stabilizable dynamical systems, in other words, controlled Markov chains. The main question here is whether a Lyapunov function in some statistical sense exists if the respective controlled Markov chain admits a stabilizing policy. We demonstrate some constructions extending on the classical results for deterministic systems. Some limitations of the constructed Lyapunov functions for stabilization are discussed, particularly for stabilization in mean. Although results for deterministic systems are well known, the stochastic case was addressed in less detail, which the current paper remarks on. A distinguishable feature of this work is the study of stabilizers that possess computationally tractable convergence certificates.