On penalisation results related with a remarkable class of submartingales

Abstract

Is this paper we study penalisations of diffusions satisfying some technical conditions, generalizing a result obtained by Najnudel, Roynette and Yor. If one of these diffusions has probability distribution $\mathbb{P}$, then our result can be described as follows: for a large class of families of probability measures $({\mathbb{Q}}_t{)}_{t\ge 0}$, each of them being absolutely continuous with respect to $\mathbb{P}$, there exists a probability ${\mathbb{Q}}_{\infty }$ such that for all events $\Lambda$ depending only on the canonical trajectory up to a fixed time, ${\mathbb{Q}}_t(\Lambda )$ tends to ${\mathbb{Q}}_{\infty }(\Lambda )$ when $t$ goes to infinity. In the cases we study here, the limit measure ${\mathbb{Q}}_{\infty }$ is absolutely continous with respect to a sigma-finite measure $\mathcal{Q}$, which does not depend on the choice of the family of probabilities $({\mathbb{Q}}_t{)}_{t\ge 0}$, but only on $\mathbb{P}$. The relation between $\mathbb{P}$ and $\mathcal{Q}$ is obtained in a very general framework by the authors of this paper.

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• From Black-Scholes and Dupire formulae to last passage times of local martingales. Part A : The infinite time horizon
• Construction of an Edwards' probability measure on $\mathcal{C}({\mathbb{R}}_{+},\mathbb{R})$