Abstracts

Our study addresses the question of how an arbitrage-free semimartingale model is affected when stopped at a random horizon or when a random variable satisfying Jacod's hypothesis is incorporated. Precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (called NUPBR hereafter) condition, which is also known in the literature as the first kind of non-arbitrage. In the general semimartingale setting, we provide a necessary and sufficient condition on the random time for which the non-arbitrage is preserved for any process. Analogous result is formulated for initial enlargement with random variable satisfying Jacod's hypothesis. The crucial intermediate results in enlargement of filtration theory are obtained. For local martingales from the reference filtration we provide special optional semimartingale decomposition up to random time and in initially enlarged filtration under Jacod's hypothesis. An interesting link to absolutely continuous change of measure problem is observed. The importance of thin random times is remarkable for our non-arbitrage considerations. In fact that is our motivation for the analysis of thin random times. We classify random times into thin and strict random times. Taking as a starting point assumption on avoidance of all stopping times from the reference filtration we define a class of thin random times. Then we define a decomposition of a random time into thin and strict parts in analogous way to the stopping time decomposition into accessible and totally inaccessible parts. The notion of dual optional projection plays a crucial role. Furthermore we develop properties of thin random times, namely relationship of thin honest times with a jumping filtration, and entropy of a thin random time.