# On penalisation results related with a remarkable class of submartingales - Mathematics > Probability

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Abstract: Is this paper we study penalisations of diffusions satisfying some technicalconditions, generalizing a result obtained by Najnudel, Roynette and Yor. Ifone of these diffusions has probability distribution $\mathbb{P}$, then ourresult can be described as follows: for a large class of families ofprobability measures $\mathbb{Q} t {t \geq 0}$, each of them being absolutelycontinuous with respect to $\mathbb{P}$, there exists a probability$\mathbb{Q} {\infty}$ such that for all events $\Lambda$ depending only on thecanonical trajectory up to a fixed time, $\mathbb{Q} t \Lambda$ tends to$\mathbb{Q} {\infty} \Lambda$ when $t$ goes to infinity. In the cases westudy here, the limit measure $\mathbb{Q} {\infty}$ is absolutely continouswith respect to a sigma-finite measure $\mathcal{Q}$, which does not depend onthe choice of the family of probabilities $\mathbb{Q} t {t \geq 0}$, but onlyon $\mathbb{P}$. The relation between $\mathbb{P}$ and $\mathcal{Q}$ isobtained in a very general framework by the authors of this paper.

Author: ** Joseph Najnudel, Ashkan Nikeghbali**

Source: https://arxiv.org/