  In biochemically reactive systems with small copy numbers of one or more
reactant molecules, the dynamics is dominated by stochastic effects. To
approximate those systems, discrete state-space and stochastic simulation
approaches have been shown to be more relevant than continuous state-space and
deterministic ones. In systems characterized by having simultaneously fast and
slow timescales, existing discrete space-state stochastic path simulation
methods, such as the stochastic simulation algorithm (SSA) and the explicit
tau-leap (Explicit-TL) method, can be very slow. Implicit approximations have
been developed to improve numerical stability and provide efficient simulation
algorithms for those systems. Here, we propose an efficient Multilevel Monte
Carlo (MLMC) method in the spirit of the work by Anderson and Higham (2012).
This method uses split-step implicit tau-leap (SSI-TL) at levels where the
SSI-TL method is not applicable due to numerical stability issues. We present
numerical examples that illustrate the performance of the proposed method.
