2009/12/24 by Joseph Najnudel, Najnudel, Joseph, Ashkan Nikeghbali +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.0912.4768
arxiv created 2009/12/24 · arxiv updated 2010/01/14
In a previous paper, we proved that for any submartingale (Xt)t ≥ 0 of class (Σ), defined on a filtered probability space (Ω, F, ℙ, (Ft)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 Λt ∈ Ft: Q [Λt, g≤ t] = 𝔼ℙ [\mathds1Λt Xt] 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.