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

Abstract

Let 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., X s, t − X 0, t − X s, 0 + X 0, 0 = ∫ Rz α( ζ, X) dB ζ + ∫ R: β( ζ, X) dζ for z ∈ R + 2, where R z = [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.