Comparison of approximate Markov generators in a one-jump setting

The problem of Markov embedding involves verifying whether a given stochastic matrix P can serve as a one-step transition matrix of a Markov chain. This is done by checking if P is the exponential of a generator matrix Q, with non-negative off-diagonal elements and zero row sums. However, it is known that a generator matrix may not be unique or may not exist.
Here, we focus on finding the approximate generator matrix under the additional assumption that the process jumps at most once during time intervals of a unit length. We determine an expression for the conditional one-step probability given at most one jump and investigate if this matrix is the same as the given transition matrix P. In this setting, we prove that for all transition matrices P with non-zero diagonal entries, the so-obtained generator matrix QJ1 is unique.
We also compare our QJ1 with the Markov generator QJLT of Jarrow, Lando, and Turnbull (1997) in their interpretation of the single jump frequency context. To this end, we study different measures of similarity between the given transition matrix P and the exponentials of QJ1 and QJLT, such as f-divergences and norms, as well as various mobility indices. We find that, in a vast number of cases, exp(QJ1) provides a closer approximation to P than exp(QJLT).