This article proposes a geostatistical solution for area-to-point spatial prediction (downscaling) taking into account boundary effects. Such effects are often poorly considered in downscaling, even though they often have significant impact on the results. The geostatistical approach proposed in this article considers two types of boundary conditions (BC), that is, a Dirichlet-type condition and a Neumann-type condition, while satisfying several critical issues in downscaling: the coherence of predictions, the explicit consideration of support differences, and the assessment of uncertainty regarding the (...) point predictions. An updating algorithm is used to reduce the computational cost of area-to-point prediction under a given BC. In a case study, area-to-point prediction under a Dirichlet-type BC and a Neumann-type BC is illustrated using simulated data, and the resulting predictions and error variances are compared with those obtained without considering such conditions. (shrink)