  We propose a general framework for studying jump-diffusion systems driven by
both Gaussian noise and a jump process with state-dependent intensity. Of
particular natural interest are the jump locations: the system evaluated at the
jump times. However, the state-dependence of the jump rate provides direct
coupling between the diffusion and jump components, making disentangling the
two to study individually difficult. We provide an iterative map formulation of
the sequence of distributions of jump locations. Computation of these
distributions allows for the extraction of the interjump time statistics. These
quantities reveal a relationship between the long-time distribution of jump
location and the stationary density of the full process. We provide a few
examples to demonstrate the analytical and numerical tools stemming from the
results proposed in the paper, including an application that shows a
non-monotonic dependence on the strength of diffusion.
