  We develop continuous time Markov chain (CTMC) approximation of
one-dimensional diffusions with a lower sticky boundary. Approximate solutions
to the action of the Feynman-Kac operator associated with a sticky diffusion
and first passage probabilities are obtained using matrix exponentials. We show
how to compute matrix exponentials efficiently and prove that a carefully
designed scheme achieves second order convergence. We also propose a scheme
based on CTMC approximation for the simulation of sticky diffusions, for which
the Euler scheme may completely fail. The efficiency of our method and its
advantages over alternative approaches are illustrated in the context of bond
pricing in a sticky short rate model for low interest environment.
