Embedding conditions for specific subsets of Markov transition matrices

Research output: Chapter in Book/Report/Conference proceedingMeeting abstract (Book)

13 Downloads (Pure)

Abstract

The initial formulation of the continuous embedding problem by Elfving in 1937 concerns the question of whether for a discrete-time Markov chain there exists a compatible continuous-time Markov process, i.e., whether for a stochastic matrix P there exists an intensity matrix R that satisfies P=e^R. If P is the estimated transition matrix based on available data on unit time intervals, stochastic matrix roots A of P contribute to insights into the transition probabilities on shorter time intervals. The discrete embedding problem addresses the question of whether for a stochastic matrix P there exists a stochastic matrix A that satisfies P = A^n for n = 2,3, .... For both cases, continuous-time and discrete-time, necessary and sufficient embedding conditions have been investigated in previous work.

Depending on the context of the system under study, the transition matrix satisfies some specific properties and, therefore, belongs to a subset S of the set П of stochastic matrices. In that case, also the matrix roots are expected to reflect these specific properties. Therefore, the aim is to find necessary and sufficient conditions for a transition matrix P\in S to have a matrix root within S. We call this the discrete embedding problem for the subset S. Because in that case the discrete embedding problem, on the one hand, concerns a subset S of stochastic matrices P, and, on the other hand, imposes the additional condition that the matrix roots A must belong to S\subsetП, it is not obvious whether the embedding conditions for the subset S are less or more restrictive than the known embedding conditions for the set П of stochastic matrices. Therefore, further investigation is needed.

For example, Markov models for credit ratings are characterized by monotone credit rating transition matrices, where each row stochastically dominates each higher row. For the specific subset S of monotone transition matrices, the discrete embedding problem and embedding conditions are presented.
Original languageEnglish
Title of host publicationProceedings SMTDA2024
PublisherISAST: International Society for the Advancement of Science and Technology.
Pages29-30
Publication statusPublished - 2024
Event8th Stochastic Modeling Techniques and Data Analysis International Conference - Cultural Center of Chania, Crete, Greece
Duration: 4 Jun 20247 Jun 2024
http://www.smtda.net/smtda2024.html

Conference

Conference8th Stochastic Modeling Techniques and Data Analysis International Conference
Abbreviated titleSMTDA 2024
Country/TerritoryGreece
CityCrete
Period4/06/247/06/24
Internet address

Keywords

  • Markov chain
  • transition matrix
  • parameter estimation
  • embedding problem
  • matrix root
  • monotone matrix

Fingerprint

Dive into the research topics of 'Embedding conditions for specific subsets of Markov transition matrices'. Together they form a unique fingerprint.

Cite this