Filtrations for the two parameter jump process

Abstract

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.Filtration stopping line two parameter process optional projection predictable projection martingale representation