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 timeinhomogeneity 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. I. MOTIVATIONWhile models describing the evolution of a set of variables are typically continuous, observations and experiments retrieve discrete sets of values. Therefore, to bridge between models and reality one has to know if it is reasonable to assume a continuous "reality" underlying the discrete set of measurements. When the evolution has a non negligible stochastic contribution, one typically extracts from the set of measurements the distribution P (X, t − τ ) of the observed values X(t − τ ), from which the probability density function (PDF) can be inferred. By knowing the distribution P (X, t) at a future time t, one is then able to define a transition matrix T(t, τ ) that satisfies:if we know the fraction of transitions T kj from each observed value X j (t − τ ) at time t − τ to a value X k (t) at time t. The transition matrix T(t, τ ) has all its elements T kj in the interval [0, 1], has row-sums one, k T kj = 1, and has non-negative entries, T kj ≥ 0.In this paper we address the problem of determining whether or not the evolution of a system is governed by a time-continuous Markov master equation. This problem is usually called the embedding problem [1]. Time-continuous Markov processes, have particular mathematical properties, namely they memoryless stochastic processes: the probability of transition between states X(t) and X(t + τ ) does not depend on the states of the system for times previous to t, for any τ > 0. If the stochastic process is time-continuous and Markovian, than the transition matrix can be defined for infinitely small τ , obeying an equation of the formwhere Q(t) is called the generator matrix of the process, having zero row-sums and non-negative off-diagonal entries. Notice that, T(t, τ ) is a transition matrix for all t and τ , i.e. with non-negative real elements and unity row-sums, if and only if it obeys Eq. (2) for some Q(t)[2]. Both equations above allow us to write the continuoustime evolution of a PDF. In other words, the...

We explicitly test if the reliability of credit ratings depends on the total number of admissible states. We analyse open access credit rating data and show that the effect of the number of states in the dynamical properties of ratings change with time, thus giving supportive evidence that the ideal number of admissible states changes with time. We use matrix estimation methods that explicitly assume the hypothesis needed for the process to be a valid rating process. By comparing with the likelihood maximization method of matrix estimation, we quantify the "likelihood-loss" of assuming that the process is a well grounded rating process.

We introduce a simple approach for testing the reliability of homogeneous generators and the Markov property of the stochastic processes underlying empirical time series of credit ratings. We analyze open access data provided by Moody's and show that the validity of these assumptions -existence of a homogeneous generator and Markovianity -is not always guaranteed. Our analysis is based on a comparison between empirical transition matrices aggregated over fixed time windows and candidate transition matrices generated from measurements taken over shorter periods. Ratings are widely used in credit risk, and are a key element in risk assessment; our results provide a tool for quantifying confidence in predictions extrapolated from rating time series.

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