The Markov chain embedding problem in a one-jump setting

Abstract

Abstract The embedding problem of Markov chains examines whether a stochastic matrix $\mathbf{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 $ \mathbf{P}=\exp{\mathbf{Q}}$ for a rate matrix $ \mathbf{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 $ \mathbf{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 $ \mathbf{P}$ . We derive a formula for this matrix in terms of $ \mathbf{Q}$ and establish that for any $ \mathbf{P}$ with non-zero diagonal entries, a unique $ \mathbf{Q}$ , called the ${\unicode{x1D7D9}}$ -generator, exists. We compare the ${\unicode{x1D7D9}}$ -generator with the one-jump rate matrix from Jarrow, Lando, and Turnbull (1997), showing which is a better approximate Markov generator of $ \mathbf{P}$ in some practical cases.