Existence and uniqueness of a strong solution to stochastic differential equations in the plane with stochastic boundary process

Let B be a 2-parameter Brownian motion on R+2. Consider the non-Markovian stochastic differential equation in the plane dX(z) = [alpha](z, X) dB(z) + [beta](z, X)dz for z [set membership, variant] R+2, i.e., Xs, t - X0, t - Xs, 0 + X0, 0 = [integral operator]Rz [alpha]([zeta], X) dB[zeta] + [integral operator]R: [beta]([zeta], X) d[zeta] for z [set membership, variant] R+2, where Rz = [0, s] - [0, t] for z = (s, t) [set membership, variant] R+2. It is shown in this paper that a unique strong solution to the stochastic differential equation exists if and only if (I) for every probability measure [mu] on the space [not partial differential]W of continuous real-valued functions on [not partial differential]R+2 there exists a solution (X, B) of the stochastic differential equation on some filtered probability space with [mu] as the probability distribution of [not partial differential]X, and (II) the pathwise uniqueness of solutions of the stochastic differential equation holds.