Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10527279 | Stochastic Processes and their Applications | 2012 | 19 Pages |
Abstract
In this paper, for any submartingale of class (Σ) defined on a filtered probability space (Ω,F,P,(Ft)tâ¥0) satisfying some technical conditions, we associate a Ï-finite measure Q on (Ω,F), such that for all tâ¥0, and for all events ÎtâFt: Q[Ît,gâ¤t]=EP[1ÎtXt], 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.
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Joseph Najnudel, Ashkan Nikeghbali,