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The abrupt changes that are ubiquitous in physical and natural systems areoften well characterized by shot noise with a state dependent recurrencefrequency and jump amplitude. For such state dependent behavior, we derive thetransition probability for both the It\^o and Stratonovich jumpinterpretations, and subsequently use the transition probability to pose amaster equation for the jump process. For exponentially distributed inputs, wepresent a novel class of transient solutions, as well as a generic steady statesolution in terms of a potential function and the Pope-Ching formula. These newresults allow us to describe state dependent jumps in a double well potentialfor steady state particle dynamics, as well as transient salinity dynamicsforced by state dependent jumps. Both examples showcase a stochasticdescription that is more general than the limiting case of Brownian motion towhich the jump process defaults in the limit of infinitely frequent and smalljumps. Accordingly, our analysis may be used to explore a continuum ofstochastic behavior from infrequent, large jumps to frequent, small jumpsapproaching a diffusion process.

State dependent jump processes: It\^o-Stratonovich interpretations, potential, and transient solutions