Predictable and dual predictable projections of two-parameter stochastic processes

In this note, we consider the existence and some basic properties of predictable and dual predictable projections of stochastic processes in the plane. The results presented here are an adaptation of an earlier version of this note which was written before the result of Bakry [1] that every two-parameter L 2 martingale possesses a c/tdlgg version. In the earlier version, predictable projections were introduced by projecting successively first with respect to the one parameter and then with respect to the other parameter. It was shown that the result is a predictable process, but it was not known wether these successive projections commute and consequently the predictable projection should not be unique. In this version, we prove the unicity of the projection following a lemma of selection for predictable sets, and the results are also based on the works of Dol6ans and Meyer [5], and Meyer [7]. Finally, we point out that the results are applicable directly to the extension of the definition of stochastic integrals in the plane.