Regularity and decomposition of two-parameter supermartingales (Q1063936)

In the first section of this article the Doob-Meyer decomposition of a supermartingale is extended in several ways to two-parameter stochastic processes. In particular the notion of Laplacian is introduced which gives a more explicit decomposition for potentials. The second section is devoted to the optional sampling theorem and to several possible notions of two-parameter local martingales. In the final section the authors give sufficient conditions for a two-parameter process to have right- continuous trajectories with limits in all the quadrants. Moreover the authors find a relation between regularity of certain quasimartingales and continuity of the associated integrable variation process.