The asymptotic behavior of the solutions of linear equations with random coefficients, random external forces and with affine boundary conditions is studied, motivated by a transmission-reflection problem for a one-dimensional wave equation in a random slab. The fluctuations of the coefficients are on a small scale in such a way that our problem is a diffusion-approximation problem except that we impose boundary conditions which force the solution to be anticipating. In the limit we obtain linear stochastic differential equations with affine boundary conditions, studied by Ocone and Pardoux. Our main tools are diffusion approximation results (Papanicolaou, Stroock and Varadhan or Ethier and Kurtz) and the properties of the limiting equations involving generalized Stratonovich integrals (Ocone and Pardoux). As an application, the transmission-reflection problem is discussed. We prove that the solution has a density with respect to the Lebesgue measure and satisfies the Markov field property.
"A Limit Theorem for Linear Boundary Value Problems in Random Media." Ann. Appl. Probab. 4 (2) 549 - 569, May, 1994. https://doi.org/10.1214/aoap/1177005072