  This paper introduces a framework for simulating finite dimensional
representations of (jump) diffusion sample paths over finite intervals, without
discretisation error (exactly), in such a way that the sample path can be
restored at any finite collection of time points. Within this framework we
extend existing exact algorithms and introduce novel adaptive approaches. We
consider an application of the methodology developed within this paper which
allows the simulation of upper and lower bounding processes which almost surely
constrain (jump) diffusion sample paths to any specified tolerance. We
demonstrate the efficacy of our approach by showing that with finite
computation it is possible to determine whether or not sample paths cross
various irregular barriers, simulate to any specified tolerance the first
hitting time of the irregular barrier and simulate killed diffusion sample
paths.
