Uniqueness of strong solutions to stochastic differential equations in the plane with deterministic boundary process (Q1103967)

Consider the stochastic differential equation of non-Markovian type in the plane \[ dX_ z=\alpha (z,X)dB_ z+\beta (z,X)dz\text{ for } z\in R^ 2_+,\quad \partial X=x, \] where B is a two-parameter Wiener process and \(\partial x\) denotes the restriction of X to the boundary \(\partial R^ 2_+\) of \(R^ 2_+\). The boundary condition x is a continuous, real-valued function on \(\partial R^ 2_+.\) The existence and uniqueness of a strong solution for this equation is obtained, assuming the existence of a weak solution and the pathwise uniqueness of solutions. The proof follows the approach of \textit{T. Yamada} and \textit{S. Watanabe}, J. Math. Kyoto Univ. 11, 155-167 (1971; Zbl 0236.60037).