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Article Dans Une Revue Electronic Journal of Probability Année : 2015

The Vervaat transform of Brownian bridges and Brownian motion

Résumé

For a continuous function f ∈ C([0, 1]), define the Vervaat transform V (f)(t) := f (τ (f) + t mod 1) + f (1)1 {t+τ (f)≥1} − f (τ (f)), where τ (f) corresponds to the first time at which the minimum of f is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.
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Dates et versions

hal-02866363 , version 1 (12-06-2020)

Identifiants

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Titus Lupu, Jim Pitman, Wenpin Tang. The Vervaat transform of Brownian bridges and Brownian motion. Electronic Journal of Probability, 2015, 20 (51), pp.1-31. ⟨10.1214/EJP.v20-3744⟩. ⟨hal-02866363⟩
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