Yan Zeng

Abstract: Let $(\Omega, ({\cal F}_t)_{t\ge 0}, {\cal F}, P)$ be a filtered probability space satisfying the usual hypotheses. Suppose $\tau$ is a stopping time and $N_t=1_{\{\tau\le t\}}$ is the associated point process. We consider when the Doob-Meyer decomposition of N has a compensator whose sample paths are a.s. absolutely continuous with respect to the Lebesgue measure. We prove a decomposition theorem which uniquely decomposes any totally inaccessible stopping time into two parts, such that one part has an absolutely continuous compensator and the other part has a singular compensator. We also characterize an absolutely continuous space, in which any totally inaccessible stopping time has an absolutely continuous compensator. A criterion of Ethier and Kurtz for the existence of an absolutely continuous compensator is extended, a corollary of which is that the absolute continuity of a compensator is preserved under filtration shrinkage.

We then define locally jumping filtrations and calculate the compensators of jump times. This result is applied to the filtration generated by a family of first passage times. Robustness of compensators is also considered in this specific setting. We then propose two formulations of filtration shrinkage and we prove they are different in general. After that, we calculate the compensators of first hitting times under various filtrations, including the natural filtration of a Hunt process or a semimartingale and the progressive expansion of shrunk filtrations. Finally, we conclude by a discussion of various computation methodologies for calculating compensators of stopping times.