  We continue the work started in Part I of this article, showing how the
addition of noise can stabilize an otherwise unstable system. The analysis
makes use of nearly optimal Lyapunov functions. In this continuation, we remove
the main limiting assumption of Part I by an inductive procedure as well as
establish a lower bound which shows that our construction is radially sharp. We
also prove a version of Peskir's \cite{Peskir_07} generalized Tanaka formula
adapted to patching together Lyapunov functions. This greatly simplifies the
analysis used in previous works.
