Sticky processes, local and true martingales

Abstract

We prove that for a so-called sticky process S there exists an equivalent probability Q and a Q-martingaleS that is arbitrarily close to S in L p (Q) norm. For continuous S,S can be chosen arbitrarily close to S in supremum norm. In the case where S is a local martingale we may choose Q arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present an application in mathematical finance. * The first author was supported by the "Lendület" Grant LP2015-6 of the Hungarian Academy of Sciences. Discussions with Martin Keller-Ressel led to formulating the main results of the present paper, we sincerely thank him. We also thank Eberhard Mayerhofer for spotting an error and two anonymous referees for helpful reports which revealed, in particular, another problem in a previous version of this paper. † MTA Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15,

Skorohod's Representation Theorem and Optimal Strategies for Markets with Frictions

“…Let us define the augmented filtration G t := F t ∨ σ(U). Standard arguments (like Lemma 4.9 of [49]) imply that G t , t ∈ [0, 1] also satisfies the usual hypotheses of completeness and right-continuity.…”