Dependence on the boundary condition for linear stochastic differential equations in the plane

An expression for the strong solution of the linear stochastic differential equation in the plane is obtained giving the solution as a function of the boundary condition. It is shown that the boundary condition as a function defined on the boundary of 2+ is transformed continuously by the solution of the stochastic differential equation as the two dimensional "time" progresses. Also the continuity of the solution jointly in 2+ and the space of boundary conditions is established.