Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² 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 ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =0, +∞)×0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . 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]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China