2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71117We prove a converse Lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions: given a stochastic system a.s. stochastic open loop stabilizable at the origin, we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium. This result provides, with the direct Lyapunov theorems proved in a companion paper, a complete Lyapunov-like characterization of the a.s. stabilizability.11 pagesOptimization and ControlProbability93E15, 49L25, 93D05text