JournalsprimsVol. 47, No. 4pp. 911–936

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

  • Joseph Najnudel

    Universität Zürich, Switzerland
  • Ashkan Nikeghbali

    Universität Zürich, Switzerland
On Some Properties of Universal Sigma-Finite Measures Associated with a Remarkable Class of Submartingales cover
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Abstract

In a previous work, we associated with any submartingale XX of class (Σ)(\Sigma), defined on a filtered probability space (Ω,F,P,(Ft)t0)(\Omega, \mathcal{F}, \mathbb{P}, (\mathcal{F}_t)_{t \geq 0}) satisfying some technical conditions, a σ\sigma-finite measure Q\mathcal{Q} on (Ω,F)(\Omega, \mathcal{F}), such that for all t0t \geq 0, and for all events ΛtFt\Lambda_t \in \mathcal{F}_t:

Q[Λt,gt]=EP[1ΛtXt],\mathcal{Q} [\Lambda_t, g\leq t] = \mathbb{E}_{\mathbb{P}} [\mathbb{1}_{\Lambda_t} X_t],

where gg is the last hitting time of zero by the process XX. 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 P\mathbb{P} with respect to a large class of functionals. The measure Q\mathcal{Q} appears to be the unifying object in these problems.

Cite this article

Joseph Najnudel, Ashkan Nikeghbali, On Some Properties of Universal Sigma-Finite Measures Associated with a Remarkable Class of Submartingales. Publ. Res. Inst. Math. Sci. 47 (2011), no. 4, pp. 911–936

DOI 10.2977/PRIMS/56