Existence and pathwise uniqueness of solutions for stochastic differential equations with respect to martingales in the plane

In this paper we establish some new theorems on pathwise uniqueness of solutions to the stochastic differential equations of the form of Xz=Z(s,0)+Z(0,t)−Z(0,0)+∫Rza(ξ,Xξ) dMξ+∫Rzb(ξ,Xξ) dAξ for z=(s,t)∈R+2 with non-Lipschitz coefficients, where M={Mz, z∈R+2} is a continuous square integrable martingale and A={Az,z∈R+2} is a continuous increasing process, Z is a continuous stochastic process on boundary ∂R+2 of R+2. We have proved existence theorem for the equation in Liang (1996a).