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Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

Bernard Roynette, Marc Yor (2010)

ESAIM: Probability and Statistics

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional: $(A_{t}^{-} : = \int_{0}^{t} 1_{X_{s} < 0} d s, t \geq 0)$ . On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [B. Roynette, P. Vallois and M. Yor, Studia Sci. Math. Hung.43 (2006) 171–246]).

Local martingales and filtration shrinkage

Hans Föllmer, Philip Protter (2011)

ESAIM: Probability and Statistics

A general theory is developed for the projection of martingale related processes onto smaller filtrations, to which they are not even adapted. Martingales, supermartingales, and semimartingales retain their nature, but the case of local martingales is more delicate, as illustrated by an explicit case study for the inverse Bessel process. This has implications for the concept of No Free Lunch with Vanishing Risk, in Finance.