Stopping times and an extension of stochastic integrals in the plane

Let ([Omega],,P;z) be a probability space with an increasing family of sub-[sigma]-fields {z, z [set membership, variant] D}, where D = [0, [infinity]) - [0, [infinity]), satisfying the usual conditions. In this paper, the stochastic integral with respect to an z-adapted 2-parameter Brownian motion for integrand processes in the class 2(z) is extended, by means of truncations cations by {0, 1}-valued 2-parameter stopping times, to integrand processes that are z-adapted and continuous. The stochastic integral in the plane thus extended resembles a locally square integrable martingale in the 1-parameter setting. A definition of a parameter-space valued, i.e., D-valued, stopping time is also given and its characteristic process is related to a {0, 1}-valued 2-parameter stopping time.