@article{01JFAGNV33ZAZAVCKBJ5QR4X8J,
  abstract     = {{The embedding problem of Markov chains examines whether a stochastic matrix P can arise as the transition matrix from time 0 to time 1 of a continuous-time Markov chain. When the chain is homogeneous, it checks if P = exp Q for a rate matrix Q with zero row sums and non-negative off-diagonal elements, called a Markov generator. It is known that a Markov generator may not always exist or be unique. This paper addresses finding Q, assuming that the process has at most one jump per unit time interval, and focuses on the problem of aligning the conditional one-jump transition matrix from time 0 to time 1 with P. We derive a formula for this matrix in terms of Q and establish that for any P with non-zero diagonal entries, a unique Q, called the 1-generator, exists. We compare the 1-generator with the one-jump rate matrix from Jarrow, Lando, and Turnbull (1997), showing which is a better approximate Markov generator of P in some practical cases.}},
  author       = {{Carette, Philippe and Guerry, Marie-Anne}},
  issn         = {{0021-9002}},
  journal      = {{JOURNAL OF APPLIED PROBABILITY}},
  keywords     = {{Markov chain,embedding problem,Markov generator,matrix exponential,BANG-BANG REPRESENTATION,TIME}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{674--696}},
  title        = {{The Markov chain embedding problem in a one-jump setting}},
  url          = {{http://doi.org/10.1017/jpr.2024.96}},
  volume       = {{62}},
  year         = {{2025}},
}

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