# Publications of the Research Institute for Mathematical Sciences

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**Volume 47, Issue 4, 2011, pp. 911–936**

**DOI: 10.2977/PRIMS/56**

Published online: 2011-11-13

On Some Properties of Universal Sigma-Finite Measures Associated with a Remarkable Class of Submartingales

Joseph Najnudel^{[1]}and Ashkan Nikeghbali

^{[2]}(1) Universität Zürich, Switzerland

(2) Universität Zürich, Switzerland

In a previous work, we associated with any submartingale $X$ of class $(\Sigma)$, defined on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0})$ satisfying some technical conditions, a $\sigma$-finite measure $\mathcal{Q}$ on $(\Omega, \mathcal{F})$, such that for all $t \geq 0$, and for all events $\Lambda_t \in \mathcal{F}_t$: $$ \mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathbb{1}_{\Lambda_t} X_t],$$ where $g$ is the last hitting time of zero by the process $X$. In this paper we establish some remarkable properties of this measure from which we also deduce a universal class of penalisation results of the probability measure $\mathbb{P}$ with respect to a large class of functionals. The measure $\mathcal{Q}$ appears to be the unifying object in these problems.

*Keywords: *martingale, submartingale, penalization

Najnudel Joseph, Nikeghbali Ashkan: On Some Properties of Universal Sigma-Finite Measures Associated with a Remarkable Class of Submartingales. *Publ. Res. Inst. Math. Sci.* 47 (2011), 911-936. doi: 10.2977/PRIMS/56