  Discrete-state, continuous-time Markov models are becoming commonplace in the
modelling of biochemical processes. The mathematical formulations that such
models lead to are opaque, and, due to their complexity, are often considered
analytically intractable. As such, a variety of Monte Carlo simulation
algorithms have been developed to explore model dynamics empirically. Whilst
well-known methods, such as the Gillespie Algorithm, can be implemented to
investigate a given model, the computational demands of traditional simulation
techniques remain a significant barrier to modern research.
  In order to further develop and explore biologically relevant stochastic
models, new and efficient computational methods are required. In this thesis,
high-performance simulation algorithms are developed to estimate summary
statistics that characterise a chosen reaction network. The algorithms make use
of variance reduction techniques, which exploit statistical properties of the
model dynamics, to improve performance.
  The multi-level method is an example of a variance reduction technique. The
method estimates summary statistics of well-mixed, spatially homogeneous models
by using estimates from multiple ensembles of sample paths of different
accuracies. In this thesis, the multi-level method is developed in three
directions: firstly, a nuanced implementation framework is described; secondly,
a reformulated method is applied to stiff reaction systems; and, finally,
different approaches to variance reduction are implemented and compared.
  The variance reduction methods that underpin the multi-level method are then
re-purposed to understand how the dynamics of a spatially-extended Markov model
are affected by changes in its input parameters. By exploiting the inherent
dynamics of spatially-extended models, an efficient finite difference scheme is
used to estimate parametric sensitivities robustly.
