  The abrupt changes that are ubiquitous in physical and natural systems are
often well characterized by shot noise with a state dependent recurrence
frequency and jump amplitude. For such state dependent behavior, we derive the
transition probability for both the It\^o and Stratonovich jump
interpretations, and subsequently use the transition probability to pose a
master equation for the jump process. For exponentially distributed inputs, we
present a novel class of transient solutions, as well as a generic steady state
solution in terms of a potential function and the Pope-Ching formula. These new
results allow us to describe state dependent jumps in a double well potential
for steady state particle dynamics, as well as transient salinity dynamics
forced by state dependent jumps. Both examples showcase a stochastic
description that is more general than the limiting case of Brownian motion to
which the jump process defaults in the limit of infinitely frequent and small
jumps. Accordingly, our analysis may be used to explore a continuum of
stochastic behavior from infrequent, large jumps to frequent, small jumps
approaching a diffusion process.
