In a previous paper, we proved that for any submartingale $(X_t)_{t \geq 0}$ of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, one can construct a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events $\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this note, we give a simpler construction of $\mathcal{Q}$, and we show that an analog of this measure can also be defined for discrete-time submartingales.

Najnudel, J; Nikeghbali, A (2010). *A new construction of the σ-finite measures associated with submartingales of class (Σ).* Comptes Rendus Mathematique, 348(5-6):311-316.

## Abstract

In a previous paper, we proved that for any submartingale $(X_t)_{t \geq 0}$ of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$, which satisfies some technical conditions, one can construct a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events $\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathds{1}_{\Lambda_t} X_t]$$ where $g$ is the last hitting time of zero of the process $X$. Some particular cases of this construction are related with Brownian penalisation or mathematical finance. In this note, we give a simpler construction of $\mathcal{Q}$, and we show that an analog of this measure can also be defined for discrete-time submartingales.

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## Additional indexing

Other titles: | Une nouvelle construction des mesures σ-finies associées aux sous-martingales de classe (Σ) |
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Item Type: | Journal Article, refereed, original work |

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | 2010 |

Deposited On: | 06 Jul 2010 12:43 |

Last Modified: | 05 Apr 2016 14:10 |

Publisher: | Elsevier |

ISSN: | 1631-073X |

Publisher DOI: | https://doi.org/10.1016/j.crma.2010.01.021 |

Related URLs: | http://arxiv.org/abs/0912.4768 |

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