Stopping times and an extension of stochastic integrals in the plane

AbstractLet (Ω,J,P;Jz) be a probability space with an increasing family of sub-σ-fields {Jz, z ∈ D}, where D = [0, ∞) × [0, ∞), satisfying the usual conditions. In this paper, the stochastic integral with respect to an Jz-adapted 2-parameter Brownian motion for integrand processes in the class C2(Jz) is extended, by means of truncations cations by {0, 1}-valued 2-parameter stopping times, to integrand processes that are Jz-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