We show how to extend a recently proposed multi-level Monte Carlo approach to
the continuous time Markov chain setting, thereby greatly lowering the
computational complexity needed to compute expected values of functions of the
state of the system to a specified accuracy. The extension is non-trivial,
exploiting a coupling of the requisite processes that is easy to simulate while
providing a small variance for the estimator. Further, and in a stark departure
from other implementations of multi-level Monte Carlo, we show how to produce
an unbiased estimator that is significantly less computationally expensive than
the usual unbiased estimator arising from exact algorithms in conjunction with
crude Monte Carlo. We thereby dramatically improve, in a quantifiable manner,
the basic computational complexity of current approaches that have many names
and variants across the scientific literature, including the
Bortz-Kalos-Lebowitz algorithm, discrete event simulation, dynamic Monte Carlo,
kinetic Monte Carlo, the n-fold way, the next reaction method,the
residence-time algorithm, the stochastic simulation algorithm, Gillespie's
algorithm, and tau-leaping. The new algorithm applies generically, but we also
give an example where the coupling idea alone, even without a multi-level
discretization, can be used to improve efficiency by exploiting system
structure. Stochastically modeled chemical reaction networks provide a very
important application for this work. Hence, we use this context for our
notation, terminology, natural scalings, and computational examples.Comment: Improved description of the constants in statement of Theorem