  We present an approach for testing for the existence of continuous generators
of discrete stochastic transition matrices. Typically, the known approaches to
ascertain the existence of continuous Markov processes are based in the
assumption that only time-homogeneous generators exist. Here, a systematic
extension to time-inhomogeneity is presented, based in new mathematical
propositions incorporating necessary and sufficient conditions, which are then
implemented computationally and applied to numerical data. A discussion
concerning the bridging between rigorous mathematical results on the existence
of generators to its computational implementation. Our detection algorithm
shows to be effective in more than $80\%$ of tested matrices, typically $90\%$
to $95\%$, and for those an estimate of the (non-homogeneous) generator matrix
follows. We also solve the embedding problem analytically for the particular
case of three-dimensional circulant matrices. Finally, a discussion of possible
applications of our framework to problems in different fields is briefly
addressed.
