Generalized product-form stationary distributions for reflected diffusions with jumps in the positive orthant

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R4 that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.