Stopping times and an extension of stochastic integrals in the plane |
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Authors: | J Yeh |
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Affiliation: | University of California, Irvine, California USA |
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Abstract: | Let (Ω,,P;z) be a probability space with an increasing family of sub-σ-fields {z, z ∈ D}, where D = [0, ∞) × [0, ∞), 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. |
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Keywords: | 60H05 Stochastic integrals in the plane 2-parameter stopping times |
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