In this paper, for any submartingale of class (Sigma) defined on a filtered probability space (Omega, F, P, (F-t)(t >= 0)) satisfying some technical conditions, we associate a sigma-finite measure Q on (Omega, F), such that for all t >= 0, and for all events Lambda(t) is an element of F-t:

Q[Lambda(t), g <= t ] = Ep[1(Lambda t)X(t)].

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. (c) 2012 Elsevier B.V. All rights reserved.

Najnudel, J; Nikeghbali, Ashkan (2012). *On some universal sigma-finite measures related to a remarkable class of submartingales.* Stochastic Processes and their Applications, 122(4):1582-1600.

## Abstract

In this paper, for any submartingale of class (Sigma) defined on a filtered probability space (Omega, F, P, (F-t)(t >= 0)) satisfying some technical conditions, we associate a sigma-finite measure Q on (Omega, F), such that for all t >= 0, and for all events Lambda(t) is an element of F-t:

Q[Lambda(t), g <= t ] = Ep[1(Lambda t)X(t)].

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. (c) 2012 Elsevier B.V. All rights reserved.

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## Additional indexing

Item Type: | Journal Article, refereed, original work |
---|---|

Communities & Collections: | 07 Faculty of Science > Institute of Mathematics |

Dewey Decimal Classification: | 510 Mathematics |

Language: | English |

Date: | April 2012 |

Deposited On: | 24 Jan 2013 14:28 |

Last Modified: | 05 Apr 2016 16:19 |

Publisher: | Elsevier |

ISSN: | 0304-4149 |

Publisher DOI: | https://doi.org/10.1016/j.spa.2012.01.010 |

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