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On some universal σ-finite measures related to a remarkable class of submartingales

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  • Najnudel, Joseph
  • Nikeghbali, Ashkan

Abstract

In this paper, for any submartingale of class (Σ) defined on a filtered probability space (Ω,F,P,(Ft)t≥0) satisfying some technical conditions, we associate a σ-finite measure Q on (Ω,F), such that for all t≥0, and for all events Λt∈Ft: Q[Λt,g≤t]=EP[1ΛtXt], where g is the last time for which the process X hits zero. The existence of Q has already been proven in several particular cases, some of them are related with Brownian penalization, and others are involved with problems in mathematical finance. More precisely, the existence of Q in the general case gives an answer to a problem stated by Madan, Roynette and Yor, in a paper about the link between the Black–Scholes formula and the last passage times of some particular submartingales. Moreover, the equality defining Q still holds if the fixed time t is replaced by any bounded stopping time. This generalization can be considered as an extension of Doob’s optional stopping theorem.

Suggested Citation

  • Najnudel, Joseph & Nikeghbali, Ashkan, 2012. "On some universal σ-finite measures related to a remarkable class of submartingales," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1582-1600.
  • Handle: RePEc:eee:spapps:v:122:y:2012:i:4:p:1582-1600
    DOI: 10.1016/j.spa.2012.01.010
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    References listed on IDEAS

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    1. Nikeghbali, Ashkan, 2006. "A class of remarkable submartingales," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 917-938, June.
    2. Laurent Carraro & Nicole El Karoui & Jan Ob{l}'oj, 2009. "On Az\'ema-Yor processes, their optimal properties and the Bachelier-drawdown equation," Papers 0902.1328, arXiv.org, revised Sep 2012.
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