  This paper is concerned with elucidating a relationship between two common
coupling methods for the continuous time Markov chain models utilized in the
cell biology literature. The couplings considered here are primarily used in a
computational framework by providing reductions in variance for different Monte
Carlo estimators, thereby allowing for significantly more accurate results for
a fixed amount of computational time. Common applications of the couplings
include the estimation of parametric sensitivities via finite difference
methods and the estimation of expectations via multi-level Monte Carlo
algorithms. While a number of coupling strategies have been proposed for the
models considered here, and a number of articles have experimentally compared
the different strategies, to date there has been no mathematical analysis
describing the connections between them. Such analyses are critical in order to
determine the best use for each. In the current paper, we show a connection
between the common reaction path (CRP) method and the split coupling (SC)
method, which is termed coupled finite differences (CFD) in the parametric
sensitivities literature. In particular, we show that the two couplings are
both limits of a third coupling strategy we call the "local-CRP" coupling, with
the split coupling method arising as a key parameter goes to infinity, and the
common reaction path coupling arising as the same parameter goes to zero. The
analysis helps explain why the split coupling method often provides a lower
variance than does the common reaction path method, a fact previously shown
experimentally.
