  This paper studies the existence and uniqueness of a classical solution to a
type of Robin boundary problems on the nonnegative orthant. We propose a new
decomposition-homogenization method for the Robin boundary problem based on
probabilistic representations, which leads to two auxiliary Robin boundary
problems admitting some simplified probabilistic representations. The auxiliary
probabilistic representations allow us to establish the existence of a unique
classical solution to the original Robin boundary problem using some stochastic
flow analysis.
