Equivalent and absolutely continuous measure changes for jump-diffusion processes

We provide explicit sufficient conditions for absolute continuity and equivalence between the distributions of two jump-diffusion processes that can explode and be killed by a potential. Introduction.The purpose of this paper is to give explicit, easy-to-check sufficient conditions for the distributions of two jump-diffusion processes to be equivalent or absolutely continuous.We consider jump-diffusions that can explode and be killed by a potential.These processes are, in general, not semimartingales.We characterize them by their infinitesimal generators.The structure of the paper is as follows.In Section 2 we introduce the notation and state the paper's main result, which gives sufficient conditions for the distributions of two jump-diffusions to be equivalent or absolutely continuous.The conditions consist of local bounds on the transformation of one generator into the other one and the assumption that the martingale problem for the second generator has for all initial distributions a unique solution.The formulation of the main theorem involves two sequences of stopping times.Stopping times of the first sequence stop the process before it explodes.The second sequence consists of exit times of the process from regions in the state space where the transformation of the first generator into the second one can be controlled.Our main result applies also in situations where the generalized Novikov condition ([19], Théorème IV.3) or Kazamaki-like criteria (e.g., [14][15][16]) are not satisfied.In Section 3 we show how X can be turned into a semimartingale by embedding it in a larger state space and stopping it before it explodes.The results of Section 3 are needed in the proof of the paper's main theorem, which is given in Section 4. In Section 5 we prove a stronger version of the result of Section 2 for a particular set-up, involving the carré-du-champ operator.In Section 6 we illustrate the main result by showing how the characteristics of a Cox-Ingersoll-Ross [3] short rate process with additional jumps and a potential can be altered by an absolutely continuous or equivalent change of measure.