2009/06/09 by Joseph Najnudel, Najnudel, Joseph, Ashkan Nikeghbali +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.0906.1782
arxiv created 2009/06/09 · arxiv updated 2009/12/01
In this paper, we associate, to any submartingale of class (Σ), defined on a filtered probability space (Ω, F, ℙ, (Ft)t ≥ 0), which satisfies some technical conditions, 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 [\mathds1Λt Xt] where g is the last hitting time of zero of the process X. This measure Q has already been defined in several particular cases, some of them are involved in the study of Brownian penalisation, and others are related with problems in mathematical finance. More precisely, the existence of Q in the general case solves a problem stated by D. Madan, B. Roynette and M. Yor, in a paper studying the link between Black-Scholes formula and last passage times of certain submartingales. Moreover, the equality defining Q remains true if one replaces the fixed time t by any bounded stopping time. This generalization can be viewed as an extension of Doob's optional stopping theorem.