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This study introduces a new method to detect continuous generators for discrete stochastic matrices, extending beyond time-homogeneous Markov processes. The algorithm effectively identifies nonhomogeneous generators and estimates them for many tested matrices.

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Area of Science:

  • Mathematics
  • Probability Theory
  • Computational Science

Background:

  • Existing methods for continuous Markov processes often assume time-homogeneous generators.
  • There is a need for methods that can handle time-inhomogeneous generators.

Purpose of the Study:

  • To develop and computationally implement a systematic approach for testing the existence of continuous generators for discrete stochastic transition matrices, including time-inhomogeneous cases.
  • To bridge the gap between theoretical mathematical results and practical computational implementation.

Main Methods:

  • Development of new mathematical propositions with necessary and sufficient conditions for generator existence.
  • Computational implementation of a detection algorithm based on these propositions.
  • Analytical solution for the embedding problem in three-dimensional circulant matrices.

Main Results:

  • The detection algorithm is effective in over 60% of tested matrices, with success rates typically between 80% and 90%.
  • For detected matrices, an estimate of the nonhomogeneous generator matrix is provided.
  • The study analytically solves the embedding problem for a specific class of matrices.

Conclusions:

  • The presented framework successfully extends the analysis of continuous Markov processes to time-inhomogeneous generators.
  • The computational approach provides a practical tool for identifying and estimating generators.
  • The framework has potential applications across various scientific fields.