 
Summary: Weak error analysis of approximate simulation methods for
multiscale stochastic chemical kinetic systems
David F. Anderson1
and Masanori Koyama2
February 14, 2011
Abstract
A chemical reaction network is a chemical system involving multiple reactions and chemical
species. The simplest stochastic models of such networks treat the system as a continuous time
Markov chain with the state being the number of molecules of each species and with reactions
modeled as possible transitions of the chain. In this paper we provide a general framework for
understanding the weak error of numerical approximation techniques in this setting. For such
models, there is typically a wide variation in scales in that the different species and reaction
rates vary over several orders of magnitude. Quantifying how different numerical approximation
techniques behave in this setting therefore requires that these scalings be taken into account
in an appropriate manner. We quantify how the error of different methods depends upon both
the natural scalings within a given system, and with the stepsize of the numerical method.
We show that Euler's method, also called explicit leaping, acts as an order one method, in
that the error decreases linearly with the stepsize, and that the approximate midpoint method
acts as either an order one or two method, depending on the relation between the timestep
and the scalings in the system. Further, we introduce a new algorithm in this setting, the weak
