On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged Filtration

Anna Aksamit (1, 2) , Tahir Choulli (3) , Monique Jeanblanc (2, 4)

1 University of Oxford
2 LaMME - Laboratoire de Mathématiques et Modélisation d'Evry
3 University of Alberta
4 UEVE - Université d'Évry-Val-d'Essonne

Given a reference filtration F, we consider the cases where an enlarged filtration G is constructed from F in two different ways: progressively with a random time or initially with a random variable. In both situations, under suitable conditions, we present a G-optional semimartingale decomposition for F-local martingales. Our study is then applied to answer the question of how an arbitrage-free semimartingale model is affected when stopped at the random time in the case of progressive enlargement or when the random variable used for initial enlargement satisfies Jacod's hypothesis. More precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition. We provide alternative proofs of some results from [5], with a methodology based on our optional semimartingale decomposition, which reduces significantly the length of the proof.

Domaines: Probabilités [math.PR]

Anna Aksamit, Tahir Choulli, Monique Jeanblanc. On an Optional Semimartingale Decomposition and the Existence of a Deflator in an Enlarged Filtration. Séminaire de Probabilités, 2015, ⟨10.1007/978-3-319-18585-9_9⟩. ⟨hal-01250968⟩