Penalisations of Brownian motion with its maximum and minimum processes as weak forms of Skorokhod embedding, X

Bernard Roynette 1, 2 Pierre Vallois 1, 2 Marc Yor 3
2 TOSCA
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We develop a Brownian penalisation procedure related to weight processes $(F_t$) of the type : $F_t := f(I_t, S_t) where $f$ is a bounded function with compact support and $S_t (resp. I_t)$ is the one-sided maximum (resp. minimum) of the Brownian motion up to time $t$. Two main cases are treated : either $F_t$ is the indicator function of ${I_t ≥ α, S_t ≤ β}$ or F_t is null when$ {S_t − I_t > c}$ for some $c > 0$. Then we apply these results to some kind of asymptotic Skorokhod embedding problem.
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Bernard Roynette, Pierre Vallois, Marc Yor. Penalisations of Brownian motion with its maximum and minimum processes as weak forms of Skorokhod embedding, X. Theory of Stochastic Processes, John Wiley & Sons, 2008, 14 (2), pp.116-138. ⟨hal-00275179⟩

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