  We 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.

Optimization and Control;Probability