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Martingales relatives

Jacques Azéma, Paul-André Meyer, Marc Yor (1992)

Séminaire de probabilités de Strasbourg

Maximal brownian motions

Jean Brossard, Michel Émery, Christophe Leuridan (2009)

Annales de l'I.H.P. Probabilités et statistiques

Let Z=(X, Y) be a planar brownian motion, 𝒵 the filtration it generates, andBa linear brownian motion in the filtration 𝒵 . One says thatB(or its filtration) is maximal if no other linear 𝒵 -brownian motion has a filtration strictly bigger than that ofB. For instance, it is shown in [In Séminaire de Probabilités XLI 265–278 (2008) Springer] that B is maximal if there exists a linear brownian motion C independent of B and such that the planar brownian motion (B, C) generates the same filtration 𝒵 asZ....

Maximal displacement for bridges of random walks in a random environment

Nina Gantert, Jonathon Peterson (2011)

Annales de l'I.H.P. Probabilités et statistiques

It is well known that the distribution of simple random walks on ℤ conditioned on returning to the origin after 2n steps does not depend on p=P(S1=1), the probability of moving to the right. Moreover, conditioned on {S2n=0} the maximal displacement maxk≤2n|Sk| converges in distribution when scaled by √n (diffusive scaling). We consider the analogous problem for transient random walks in random environments on ℤ. We show that under the quenched law Pω (conditioned on the environment ω), the maximal...

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