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

Bernard Roynette (1, 2) , Pierre P. Vallois (1, 2) , Marc Yor (3)

1 IECN - Institut Élie Cartan de Nancy
2 TOSCA - Simuler et calibrer des modèles stochastiques
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 ${I_t\geq \alpha, S_t \leq \beta}$ 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.

Mots clés: Skorokhod's problem; penalisation; one-sided maximum and minimum; Laplace's method

Domaines: Probabilités [math.PR]

Bernard Roynette, Pierre P. Vallois, Marc Yor. Penalisations of Brownian motion with its maximum and minimum processes as weak forms of Skorokhod embedding, X. Theory of Stochastic Processes, 2008, 14 (2), pp.116-138. ⟨hal-00275179⟩