Summary: An Efficient Finite Difference Method for Parameter Sensitivities of
Continuous Time Markov Chains
David F. Anderson1
September 13, 2011
We present an efficient finite difference method for the computation of parameter sensitivities for a
wide class of continuous time Markov chains. The motivating class of models, and the source of our
examples, are the stochastic chemical kinetic models commonly used in the biosciences, though other
natural application areas include population processes and queuing networks. The method is essentially
derived by making effective use of the random time change representation of Kurtz, and is no harder to
implement than any standard continuous time Markov chain algorithm, such as "Gillespie's algorithm"
or the next reaction method. Further, the method is analytically tractable, and, for a given number of
realizations of the stochastic process, produces an estimator with substantially lower variance than that
obtained using other common methods. Therefore, the computational complexity required to solve a
given problem is lowered greatly. In this work, we present the method together with the theoretical
analysis detailing the variance of the resulting estimator of the sensitivities. We also provide numerical
examples comparing the method developed here to other common methods.
Keywords: finite difference, variance reduction, parameter sensitivities, next reaction method, random time
change, continuous time Markov chain.