A new construction of the σ-finite measures associated with submartingales of class (Σ)
Najnudel, Joseph · Nikeghbali, Ashkan
Original · EN
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 [1Λₜ 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.
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