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

AbstractLet 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 ΔtM = lims↑t[Mt − M(t1,s2) − M(s1,t2) + Ms] ≠ 0, a 1-jump if ΔtM = 0 and lims1↑t1[Mt − M(s1,t2)] ≠ 0. Analogously, 2-jumps are defined. With the 0-jumps associate the two-parameter point process μM which assigns unit point mass to nontrivial (t, ΔtM), with the 1-jumps the one-parameter point process μ1M which puts unit mass to nontrivial (t1, Δt1M(+, 1)), and with the 2-jumps a corresponding μ2M. We define stochastic integrals with respect to the compensated μM, μiM, 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