Lifting theorems for some clases of two parameter martingales

Abstract

By means of nonstandard analysis we establish some lifting theorerms for two parameter stochastic processes, for two parameter martingales and for weak, strong and i-martingales. We also prove that the standard part of an internal martingale is a standard larc martingale (a two parameter version of a cadlag martingale). A basic nonstandard two parameter stochastic integral is introduced. An integral representation of Wong and Zakai proves to be a very useful tool for our purposes.