The effect of the number of states on the validity of credit ratings

Lencastre, Pedro; Raischel, Frank; Lind, Pedro G.

doi:10.1088/1742-6596/574/1/012151

Cited by 1 publication

(5 citation statements)

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“…This problem as already addressed by us [6,7] in the particular case of homogeneous transition matrices derived by rating agencies. Further, our methodology could be extended to other situations where correlation matrices are taken for describing the macroscopic state of a financial system [25].…”

Section: Discussionmentioning

confidence: 91%

“…One important interdisciplinary application is, of course, in economics and finance, when addressing rating matrices: if ratings do indeed reflect a natural (continuous) economic process, the extracted rating matrices must have a proper generator [24] . This problem as already addressed by us [6,7] in the particular case of homogeneous transition matrices derived by rating agencies. Further, our methodology could be extended to other situations where correlation matrices are taken for describing the macroscopic state of a financial system [25].…”

Section: Discussionmentioning

confidence: 92%

“…In this paper we dealt typically with type (ii) data sets, while in previous works [6,7] we considered mainly type (iii). Type (i) is typically not well-defined and additional cautions must be taken.…”

Section: Discussionmentioning

confidence: 99%

“…For specific applications, our framework can be used for three types of stochastic data sets: (i) one where only the initial and final configuration of the system is given; (ii) one where all possible state transitions are defined through a probability value between the start and end of the observation period and (iii) the transition between the beginning of intermediate instants till the end of the observation. In this paper we dealt typically with type (ii) data sets, while in previous works [6,7] we considered mainly type (iii). Type (i) is typically not well-defined and additional cautions must be taken.…”

Section: Discussionmentioning

confidence: 99%

“…The first one is when the underlying process is not Markovian. Such scenario was previously addressed by us [6,7]. One second reason is the statistical error any empirical data set is subjected to.…”

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confidence: 99%

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From empirical data to time-inhomogeneous continuous Markov processes

et al. 2016

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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...

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Section: Discussionmentioning

confidence: 91%

Section: Discussionmentioning

confidence: 92%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

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