ABSTRACT
In this chapter, the authors state and prove a large deviations principle for a class of multidimensional jump processes. The idea behind the proof of Kurtz's Theorem is very similar to the idea behind large deviations upper bounds: in each direction , we estimate how much can differ from its mean by using an exponential martingale (the process version of Chebycheff's inequality). Paradoxically, the upper bound can be established (at the present time) in greater generality than the lower bound. The first key observation is that, with exponentially high probability, the processes stay in compact sets: in the nomenclature of large deviations, is exponentially tight. The chapter explains that the Kurtz’s Theorem is an extension of the law of large numbers. The proof of Kurtz’s Theorem is presented as a series of lemmas. The chapter states the technical results, then provide an outline of the proof of Kurtz’s Theorem, and then proceed with the proof.
