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A class of recursive filtering problems for random fields with a two-dimensional parameter is considered. After a brief introduction of two-parameter stochastic calculus, a class of Markovian random fields generated by stochastic integral equations is defined and considered. It is then shown that the problem of estimating such a Markovian field in additive white Gaussian noise can be reduced to a recursive formalism. If the random field is itself Gaussian, the recursive formalism reduces to a finite set of stochastic integral equations involving the conditional mean and covariance.