  The existing literature on stochastic simulation of chemical reaction
networks has a tendency to move as quickly as possible to the abstract
formulation of the stochastic dynamics in terms of probabilities based on the
concept of the Chemical Master Equation (CME), largely ignoring sample path
representation.
  In this publication we discuss both theoretical basis and numerical approach
for the problems in this area using sample path methods as a crucial part of
the process. Relying as it does on a representation of the underlying
stochastic processes as a weak solution of a system of stochastic differential
equations driven by Poisson random measures this approach brings to bear a
heretofore ignored but quite effective problem solving methodology. We first
present a simple and intuitive way of partitioning species and reactions of the
interaction network into different groups. We then discuss how original
stochastic dynamics with state dependent intensities of transitions can be
reformulated in terms of jump-diffusion stochastic differential equations
driven by both Wiener noise sources and Poisson random measures. Finally, we
show that this approach facilitates the construction of hybrid simulation
techniques, an important step in the creation of efficient techniques for
modeling multi-scale stochastic dynamics of the reaction networks.
  Numerical methods related to sampling events from Poisson random measures are
demonstratedon simple intuitive examples. Error control analysis of the finite
differences scheme is also presented.
