Markov Chain Approximation Algorithm EMCEL, Its Convergence and Perturbation Analysis

We propose a new algorithm, called EMCEL (Embeddable Markov Chain with Expected time Lag h), for approximating continuous Markov processes in law. The approximation is a discrete-time Markov chain, which in many cases can be constructed explicitly. We prove that the algorithm applies to all one-dimensional continuous strong Markov processes. E.g., superlinear growth of the diffusion coefficient, presence of sticky points and other possible irregularities do not destroy the convergence. We also perform a perturbation analysis for EMCEL, that is, in the cases when the construction of the Markov chains prescribed by the algorithm cannot be performed exactly and we thus need to ``approximate the approximating Markov chains'' (this second approximation step is deterministic), we analyze how small the error in the latter approximation step should be in order to retain the convergence. The algorithm is illustrated by a couple of examples, including sticky Brownian motion and a Brownian motion slowed down on the Cantor set.

This is a joint work with Stefan Ankirchner and Thomas Kruse.