The martingale theory of jump processes

A jump process is best analyzed by investigating the space of all martingales which are generated by the process. The structure of this space of martingales becomes clear in the martingale representation results. Having understood this structure one can completely resolve the most important problems in i) modelling and description, ii) detection or hypothesis testing and iii) filtering of jump processes. The discussion here is not the most general possible, but it covers all the important practical situations.