Explicit form and robustness of martingale representations

Stochastic integral representation of martingales has been undergoing a ren aissance due to questions motivated by stochastic finance theory. In the Br ownian case one usually has formulas (of differing degrees of exactness) fo r the predictable integrands. We extend some of these to Markov cases where one does not necessarily have stochastic integral representation of all ma rtingales. Moreover we study various convergence questions that arise natur ally from (for example) approximations of "price processes" via Euler schem es for solutions of stochastic differential equations. We obtain general re sults of the following type: let U, U-n be random variables with decomposit ions U = alpha + integral (infinity)(0) xi (s) dX(s) +N-infinity, U-n = alpha (n) + integral (infinity)(0) xi (n)(s) dX(s)(n) + N-infinity(n) , where X, N, X-n, N-n are martingales. If X-n --> X and U-n --> U, when and how does xi (n) --> xi?