Abstract
In Najnudel and Nikeghbali (2009) [7], we prove that for any submartingale ( X t ) t ⩾ 0 of class ( Σ), defined on a filtered probability space ( Ω , F , P , ( F t ) 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 Λ t ∈ F t : Q [ Λ t , g ⩽ t ] = E P [ 1 Λ t X t ] 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.
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