Stochastic integral representation of martingales has been
undergoing a renaissance due to questions motivated by stochastic finance
theory. In the Brownian case one usually has formulas (of differing degrees of
exactness) for the predictable integrands. We extend some of these to Markov
cases where one does not necessarily have stochastic integral representation of
all martingales. Moreover we study various convergence questions that arise
naturally from (for example)approximations of “price processes”
via Euler schemes for solutions of stochastic differential equations. We obtain
general results of the following type: let $U, U^n$ be random variables with
decompositions
$$ U = \alpha + \int_{0}^{\infty} \xi_s dX_s +
N_\infty, $$
$$ U^n = \alpha_n + \int_{0}^{\infty} \xi^n_s dX^n_s
+ N^n_\infty, $$
where $X, N, X^n, N^n$ are martingales. If $X^n \to X$
and $U^n \to U$, when and how does $\xi^n\rightarrow\xi?$