  Motivated by the lack of a suitable constructive framework for analyzing
popular stochastic models of Systems Biology, we devise conditions for
existence and uniqueness of solutions to certain jump stochastic differential
equations (SDEs). Working from simple examples we find reasonable and explicit
assumptions on the driving coefficients for the SDE representation to make
sense. By `reasonable' we mean that stronger assumptions generally do not hold
for systems of practical interest. In particular, we argue against the
traditional use of global Lipschitz conditions and certain common growth
restrictions. By `explicit', finally, we like to highlight the fact that the
various constants occurring among our assumptions all can be determined once
the model is fixed.
  We show how basic long time estimates and some limit results for
perturbations can be derived in this setting such that these can be contrasted
with the corresponding estimates from deterministic dynamics. The main
complication is that the natural path-wise representation is generated by a
counting measure with an intensity that depends nonlinearly on the state.
