# A new construction of the σ-finite measures associated with submartingales of class (Σ) - Zurich Open Repository and Archive

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.

## 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.

## Citations

1 citation in Web of Science®
2 citations in Scopus®

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Other titles: Une nouvelle construction des mesures σ-finies associées aux sous-martingales de classe (Σ) Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2010 06 Jul 2010 12:43 05 Apr 2016 14:10 Elsevier 1631-073X https://doi.org/10.1016/j.crma.2010.01.021 http://arxiv.org/abs/0912.4768