Explicit form and robustness of martingale representations

Abstract

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 $\xi^n\rightarrow\xi?$