In this paper we specify the basic set of economic criteria that any diffusion-driven interest rate or FX rate process must satisfy. We also develop the methodology that is implementable to test the validity of a proposed process insofar as it satisfies the basic criteria as well as the actual estimation of the parameters of an acceptable candidate process. In this paper we focus on processes such as the overnight repo rate process or the FX rate process, each of which is directly observable. We develop what we call the marginal maximum-likelihood estimation (MMLE) technique to distinguish it from the joint maximum-likelihood estimation (JMLE) technique, which we present in a separate paper. We also present some preliminary empirical results for both the interest rate process and the FX rate process.