Converse theorems for stochastic exponential stability

If a continuous strong Markov process is exponentially stable w.p.l., then there exists a Lyapunov function which satisfies an exponential inequality along the process. An analogous result is also true for discrete-time processes, but requires a different construction for the Lyapunov function. This is in contrast to the case of asymptotic stability w.p.l. for which the same construction is valid for both continuous and discrete-time processes.