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

AbstractLet B be a 2-parameter Brownian motion on R+2. Consider the non-Markovian stochastic differential equation in the plane dX(z) = α(z, X) dB(z) + β(z, X)dz for z ∈ R+2, i.e., Xs, t − X0, t − Xs, 0 + X0, 0 = ∫Rz α(ζ, X) dBζ + ∫R: β(ζ, X) dζ for z ∈ R+2, where Rz = [0, s] × [0, t] for z = (s, t) ∈ 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 μ on the space ∂W of continuous real-valued functions on ∂R+2 there exists a solution (X, B) of the stochastic differential equation on some filtered probability space with μ as the probability distribution of ∂X, and (II) the pathwise uniqueness of solutions of the stochastic differential equation holds