A new construction of the <i>σ</i>-finite measures associated with submartingales of class (<i>Σ</i>)

Joseph Najnudel; Ashkan Nikeghbali

doi:10.1016/j.crma.2010.01.021

Probability Theory

A new construction of the σ-finite measures associated with submartingales of class (Σ)
[Une nouvelle construction des mesures σ-finies associées aux sous-martingales de classe (Σ)]

Présenté par : Marc Yor

Joseph Najnudel ¹ ; Ashkan Nikeghbali ¹

¹ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 311-316.

Résumé
Abstract

Dans Najnudel et Nikeghbali (2009) [7], nous prouvons que pour toute sous-martingale ${(X_{t})}_{t ⩾ 0}$ de classe (Σ), définie sur un espace de probabilité filtré $(Ω, F, P, {(F_{t})}_{t ⩾ 0})$ , satisfaisant certaines conditions techniques, on peut construire une mesure σ-finie $Q$ sur $(Ω, F)$ , telle que pour tout $t ⩾ 0$ , et pour tout événement $Λ_{t} \in F_{t}$ :

Q [Λ_{t}, g ⩽ t] = E_{P} [1_{Λ_{t}} X_{t}]

où g est le dernier zéro de X. Certains cas particuliers de cette construction sont liés aux pénalisations browniennes ou aux mathématiques financières. Dans cette note, nous donnons une construction plus simple de

Q

, et nous montrons qu'un analogue de cette mesure peut aussi être défini pour des sous-martingales à temps discret.

In Najnudel and Nikeghbali (2009) [7], we prove that for any submartingale ${(X_{t})}_{t ⩾ 0}$ of class (Σ), defined on a filtered probability space $(Ω, F, P, {(F_{t})}_{t ⩾ 0})$ , which satisfies some technical conditions, one can construct a σ-finite measure $Q$ on $(Ω, F)$ , such that for all $t ⩾ 0$ , and for all events $Λ_{t} \in F_{t}$ :

Q [Λ_{t}, g ⩽ t] = E_{P} [1_{Λ_{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

Q

, and we show that an analog of this measure can also be defined for discrete-time submartingales.

Reçu le : 2009-12-23
Accepté le : 2010-01-12
Publié le : 2010-02-26

DOI : 10.1016/j.crma.2010.01.021

Affiliations des auteurs :

Joseph Najnudel ¹ ; Ashkan Nikeghbali ¹

¹ Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

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     author = {Joseph Najnudel and Ashkan Nikeghbali},
     title = {A new construction of the \protect\emph{\ensuremath{\sigma}}-finite measures associated with submartingales of class {(\protect\emph{\ensuremath{\Sigma}})}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {311--316},
     publisher = {Elsevier},
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     year = {2010},
     doi = {10.1016/j.crma.2010.01.021},
     language = {en},
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Joseph Najnudel; Ashkan Nikeghbali. A new construction of the σ-finite measures associated with submartingales of class (Σ). Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 311-316. doi : 10.1016/j.crma.2010.01.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.021/

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