Computing <i>p</i> th root of a transition matrix with a deep unsupervised learning approach

Transition matrices are a tool used to describe transition probabilities of a Markovian process. From an applied perspective, transition matrices are, for example, used in the fields of credit risk modeling and medical decision analysis. In these domains, it is possible to estimate matrices according to the frequency of data observations, but for modeling purposes, it may be necessary to calculate transition matrices for shorter and more restricted time intervals. From a mathematical perspective, this requirement translates into calculating the pth root of a stochastic matrix through a constrained optimization problem. This article focuses on computing pth root of transition matrices and proposes a novel approach using deep learning-based algorithms, particularly Convolutional Neural Networks. Comparisons with traditional algorithms reveal that, in certain contexts, deep learning models offer computational advantages, especially regarding time efficiency. The applicability of the models is verified through a case study built on Fitch Ratings data.