Stopping times and an extension of stochastic integrals in the plane

Let (Ω,$\text{J}$,P;$\text{J}$_z) be a probability space with an increasing family of sub-σ-fields {$\text{J}$_z, z ∈ D}, where D = [0, ∞) × [0, ∞), satisfying the usual conditions. In this paper, the stochastic integral with respect to an $\text{J}$_z-adapted 2-parameter Brownian motion for integrand processes in the class $\text{C}$₂($\text{J}$_z) is extended, by means of truncations cations by {0, 1}-valued 2-parameter stopping times, to integrand processes that are $\text{J}$_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.