Stationarity in Markovian Marked Point Process and Jump-Diffusion Models
Master's Thesis for degree in Mathematics
Many phenomena exhibit a piecewise continuous non-deterministic behavior. Jump-diffusions with jumps from a renewal process are an obvious choice for modeling such behavior. The key question treated in this thesis is to investigate when a stationary distribution exists and, if possible, to find it.
Two different models are considered. The first specifies the target just after each jump, while the second specifies the change relative to the position of the process just before the jump. Applications of both models are briefly touched upon.
The existence and uniqueness of a stationary distribution is shown using the key renewal theorem for a very broad class of models. In case of the jump target model, renewal theory offers a way to find the stationary distribution while a integro-differential equation, sometimes solvable, for the density is also presented.
The jump size model is harder to deal with. In certain cases it is possible to reduce the analysis to a jump target model. Nevertheless, explicitness is much harder to attain.
For both models, results are more tangible when the jumps stem from a Poisson process. For such models, in the jump target case we are able to find the stationary density for many popular diffusion choices.