Uniqueness theorem of solutions for stochastic differential equation in the plane.

Let M={M _z, z ∈ R} be a continuous square integrable martingale and A={A _z, z ∈ R be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane: dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R, X_z=Z_z, z∈∂R, where R =[0, +∞)×[0,+∞) and ∂ R is its boundary, Z is a continuous stochastic process on ∂ R . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. [ABSTRACT FROM AUTHOR]