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The embedding problem of Markov chains is a long standing problem where a given stochastic matrix is examined as the 1-step transition matrix of some continuous-time homogeneous Markov chain (CTHMC). This problem boils down to characterizing the empirical transition matrix P as the exponential of some matrix Q with all non-negative off-diagonal entries and zero row-sums, called an intensity matrix. If such a Q exists, P is said to be embeddable. It turns out that the embedding problem is a formidable one in a number of respects. First, P may not be embeddable. In that case, a regularization algorithm can be used to find an intensity matrix Q for which ||P - exp(Q)|| is minimized. Next, no embeddability criteria in terms of the matrix elements, which are easily verifiable in practice, seem at hand when the number of states exceeds 3 . Lastly, for an embeddable P, there may not be a unique solution to the equation exp(Q) = P in the set of intensity matrices. The identification aspect of the embedding problem deals with the selection of the suitable intensity matrix reflecting the nature of the system under study. We propose the conditional embedding approach where the empirical 1-step transition matrix P corresponds with the conditional 1-step transition matrix of the CTHMC given the event that at most one jump has occurred during a time interval of unit length. For a Markov model the unit time interval can be defined in such a way that the empirical 1-step transition matrix meets this condition. Moreover, this condition is inherent in some applications. For example, in credit rating migration models the credit ratings are typically based on slowly varying characteristics, such that they do not tend to change more than once within the baseline time interval (e.g. a quarter). We found that, regardless the number of states, exactly one intensity matrix solves this conditional embedding problem when Pii > 0 for all i. Our approach results in an easy embeddability criterium and does not require identification neither regularization. Keywords: Embedding problem; Markov chain; transition matrix.

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MLA
Carette, Philippe, and Marie-Anne Guerry. “A Conditional Embedding Problem for Finite Homogeneous Markov Chains.” ASMDA2023, 20th Applied Stochastic Models and Data Analysis International Conference with Demographics Workshop, Book of Abstracts, edited by Christos H. Skiadas, International Society for the Advancement of Science and Technology (ISAST), 2023, pp. 24–25.
APA
Carette, P., & Guerry, M.-A. (2023). A conditional embedding problem for finite homogeneous Markov chains. In C. H. Skiadas (Ed.), ASMDA2023, 20th Applied Stochastic Models and Data Analysis International Conference with Demographics Workshop, Book of abstracts (pp. 24–25). International Society for the Advancement of Science and Technology (ISAST).
Chicago author-date
Carette, Philippe, and Marie-Anne Guerry. 2023. “A Conditional Embedding Problem for Finite Homogeneous Markov Chains.” In ASMDA2023, 20th Applied Stochastic Models and Data Analysis International Conference with Demographics Workshop, Book of Abstracts, edited by Christos H. Skiadas, 24–25. International Society for the Advancement of Science and Technology (ISAST).
Chicago author-date (all authors)
Carette, Philippe, and Marie-Anne Guerry. 2023. “A Conditional Embedding Problem for Finite Homogeneous Markov Chains.” In ASMDA2023, 20th Applied Stochastic Models and Data Analysis International Conference with Demographics Workshop, Book of Abstracts, ed by. Christos H. Skiadas, 24–25. International Society for the Advancement of Science and Technology (ISAST).
Vancouver
1.
Carette P, Guerry M-A. A conditional embedding problem for finite homogeneous Markov chains. In: Skiadas CH, editor. ASMDA2023, 20th Applied Stochastic Models and Data Analysis International Conference with Demographics Workshop, Book of abstracts. International Society for the Advancement of Science and Technology (ISAST); 2023. p. 24–5.
IEEE
[1]
P. Carette and M.-A. Guerry, “A conditional embedding problem for finite homogeneous Markov chains,” in ASMDA2023, 20th Applied Stochastic Models and Data Analysis International Conference with Demographics Workshop, Book of abstracts, Heraklion, Crete, Greece, 2023, pp. 24–25.
@inproceedings{01H27Z6JPAMT43S8DCT40JTASK,
  abstract     = {{The embedding problem of Markov chains is a long standing problem  where a given stochastic matrix is examined as the 1-step transition matrix of some continuous-time homogeneous Markov chain (CTHMC). This problem boils down to characterizing the empirical transition matrix P as the exponential of some matrix Q with all non-negative off-diagonal entries and zero row-sums, called an intensity matrix. If such a Q exists, P is said to be embeddable. It turns out that the embedding problem is a formidable one in a number of respects. First, P may not be embeddable. In that case, a regularization algorithm can be used to find an intensity matrix Q for which ||P - exp(Q)|| is minimized. Next, no embeddability criteria in terms of the matrix elements, which are easily verifiable in practice, seem at hand when the number of states exceeds 3 . Lastly, for an embeddable P, there may not be a unique solution to the equation exp(Q) = P in the set of intensity matrices. The identification aspect of the embedding problem deals with the selection of the suitable intensity matrix reflecting the nature of the system under study.
We propose the conditional embedding approach where the empirical 1-step transition matrix P corresponds with the conditional 1-step transition matrix  of the CTHMC given the event that at most one jump has occurred during a time interval of unit length. For a Markov model the unit time interval can be defined in such a way that the empirical 1-step transition matrix meets this condition. Moreover, this condition is inherent in some applications. For example, in credit rating migration models the credit ratings are typically based on slowly varying characteristics, such that they do not tend to change more than once within the baseline time interval (e.g. a quarter).
We found that, regardless the number of states, exactly one intensity matrix solves this conditional embedding problem when Pii > 0 for all i. Our approach results in an easy embeddability criterium and does not require identification neither regularization.
Keywords: Embedding problem; Markov chain; transition matrix.}},
  author       = {{Carette, Philippe and Guerry, Marie-Anne}},
  booktitle    = {{ASMDA2023, 20th Applied Stochastic Models and Data Analysis International Conference with Demographics Workshop, Book of abstracts}},
  editor       = {{Skiadas, Christos H.}},
  language     = {{eng}},
  location     = {{Heraklion, Crete, Greece}},
  pages        = {{24--25}},
  publisher    = {{International Society for the Advancement of Science and Technology (ISAST)}},
  title        = {{A conditional embedding problem for finite homogeneous Markov chains}},
  url          = {{www.asmda.es}},
  year         = {{2023}},
}