Stochastic integrals of point processes and the decomposition of two-parameter martingales

Let M be a square integrable martingale indexed by [0, 1]2 with respect to a filtration which possesses the property of conditional independence. Assume that M has trajectories which are continuous for approach from the right upper quadrant and possess limits for the remaining three. M can have three kinds of jumps. A point t is a 0-jump if [Delta]tM = lims[short up arrow]t[Mt - M(t1,s2) - M(s1,t2) + Ms] [not equal to] 0, a 1-jump if [Delta]tM = 0 and lims1[short up arrow]t1[Mt - M(s1,t2)] [not equal to] 0. Analogously, 2-jumps are defined. With the 0-jumps associate the two-parameter point process [mu]M which assigns unit point mass to nontrivial (t, [Delta]tM), with the 1-jumps the one-parameter point process [mu]1M which puts unit mass to nontrivial (t1, [Delta]t1M(+, 1)), and with the 2-jumps a corresponding [mu]2M. We define stochastic integrals with respect to the compensated [mu]M, [mu]iM, i = 1, 2, with the help of which we can describe the jump components associated with the respective jumps in the orthogonal decomposition of M by discontinuous and continuous parts.