Brownian penalisations related to excursion lengths, VII

Applications of the continuous - time ballot theorem to Brownian motion and related processes

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to rst hit > 0 at tim...

Cameron–Martin formula for the σ-finite measure unifying Brownian penalisations

Quasi-invariance under translation is established for the σ-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor ([10]). For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor ([5]), plays a key role.

Wiener integral for the coordinate process under the σ-finite measure unifying Brownian penalisations

Wiener integral for the coordinate process is defined under the σ-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor ([8] and [9]). Its decomposition before and after last exit time from 0 is studied. This study prepares for the author’s recent study [12] of Cameron–Martin formula for the σ-finite measure.

Penalisations of multidimensional Brownian motion, VI

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

We develop a Brownian penalisation procedure related to weight processes (Ft) of the type : Ft := f(It, St) where f is a bounded function with compact support and St (resp. It) is the one-sided maximum (resp. minimum) of the Brownian motion up to time t. Two main cases are treated : either Ft is the indicator function of {It ≥ α, St ≤ β} or Ft is null when {St − It > c} for some c > 0. Then we ...