Filtrations for the two parameter jump process

A process which has just one jump, and whose time parameter is the positive quadrant 0, ∞] × 0, ∞], is considered. Following Merzbach, related stopping lines are introduced, and the filtration {$\text{F}$_t1,t2³} considered in this paper is such that, modulo completion, the σ-field $\text{F}$_t1,t2³ is the Borel field on the region

$\text{L}{}_{\mathrm{t1,t2}}\text{={(s}{}_1\text{,s}{}_2\text{); 0?s}{}_1\text{?t}{}_1\text{or}\text{0?s}{}_2\text{?t}{}_2\text{}}$

, together with the atom which is the complement in Ω = 0, ∞]² of L_t1,t2. Optional and predictable projections of related processes are defined, together with their dual projections, and an integral representation for martingales is obtained.