The concept of a semi-martingale is extended to processes with index set in the plane. The definitions of planar semi-martingales are similar to those of two parameter bounded variation. Necessary and sufficient conditions for a Doob-Meyer decomposition are obtained, and a maximal inequality and...

Stochastic integration of left continuous integrands with respect to quasimartingales is developed as the pathwise limit of Riemann-Stieltjes sums. The procedure is extended to right continuous integrands.