Extending the multi-level method for the simulation of stochastic biological systems

The multi-level method for discrete state systems, first introduced by Anderson
and Higham (2012), is a highly efficient simulation technique that can be
used to elucidate statistical characteristics of biochemical reaction networks. A single point estimator is produced in a cost-effective manner by combining a number of estimators of differing accuracy in a telescoping sum, and, as such, the method has the potential to revolutionise the field of stochastic simulation. The first term in the sum is calculated using an approximate simulation algorithm, and can be calculated quickly but is of significant bias. Subsequent terms successively correct this bias by combining estimators from approximate stochastic simulations algorithms of increasing accuracy, until a desired level of accuracy is reached.
In this paper we present several refinements of the multi-level method which render it easier to understand and implement, and also more efficient. Given the substantial and complex nature of the multi-level method, the first part of this work reviews existing literature, with the aim of providing a practical guide to the use of the multilevel method. The second part provides the means for a deft implementation of the technique, and concludes with a discussion of a number of open problems.