Filtrations for the two parameter jump process

A process which has just one jump, and whose time parameter is the positive quadrant [0, [infinity]] - [0, [infinity]], is considered. Following Merzbach, related stopping lines are introduced, and the filtration {t1,t23} considered in this paper is such that, modulo completion, the [sigma]-field t1,t23 is the Borel field on the region Lt1,t2={(s1,s2); 0[less-than-or-equals, slant]s1[less-than-or-equals, slant]t1or0[less-than-or-equals, slant]s2[less-than-or-equals, slant]t2}, together with the atom which is the complement in [Omega] = [0, [infinity]]2 of Lt1,t2. Optional and predictable projections of related processes are defined, together with their dual projections, and an integral representation for martingales is obtained.