Markov chain approximation of one-dimensional singular diffusions

这个项目的目的是构造一个弱逼近方案，它适用于一大类可能具有奇异行为的一维连续强马氏过程。例如，应包括域内粘性或(瞬时或缓慢)边界反射等现象。近似应该是可在计算机上实现的马尔可夫链。我们对路径泛函的近似很感兴趣。因此，应该推导出适合我们目的的泛函极限定理。由于在实际应用中经常会出现逼近某些不连续路径泛函(如退出域的时间)的问题，我们的目标是构造一个能够做到这一点的方案。还应研究该格式的主要性质，包括它的收敛速度，以及它在重要例子上的性能。

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.