Markov chain approximation of one-dimensional singular diffusions

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The aim of this project is to construct a weak approximation scheme that applies to large classes of one-dimensional continuous strong Markov processes with possibly singular behaviour. E.g. such phenomena as stickiness inside the domain or (instanteneous or slow) reflection at the boundary should be included. Approximations should be Markov chains implementable on a computer. We are interested in approximations of path functionals. Therefore, an appropriate for our purpose functional limit theorem should be derived. Since the question of approximating certain discontinuous path functionals (such as exit times from the domain) often arises in practice, we aim at constructing a scheme capable of doing that. The main properties of the scheme, including its convergence rate, and its performance on important examples should be studied as well.