Weak convergence of semimartingales and discretisation methods

Given a semimartingale one can construct a system ([lambda], A, B, C) where [lambda] is the distribution of the initial value and (A, B, C) is the triple of global characteristics. Thus, given a process X and a system ([lambda], A, B, C) one can look for all probability measures P such that X is a P-semimartingale with initial distribution [lambda] and global characteristics (A, B, C). We say that such a measure P is a solution to the semimartingale problem ([lambda], A, B, C). The paper is devoted to the study of a special type of semimartingale problem. We look for sufficient conditions to insure the existence of solutions and we develop a method to construct them by means of time-discretised schemes, using weak topology for probability measures.