  We develop a direct Lyapunov method for the almost sure open-loop
stabilizability and asymptotic stabilizability of controlled degenerate
diffusion processes. The infinitesimal decrease condition for a Lyapunov
function is a new form of Hamilton-Jacobi-Bellman partial differential
inequality of $2nd$ order. We give local and global versions of the First and
Second Lyapunov Theorems assuming the existence of a lower semicontinuous
Lyapunov function satisfying such inequality in the viscosity sense. An
explicit formula for a stabilizing feedback is provided for affine systems with
smooth Lyapunov function. Several examples illustrate the theory.
