Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Moment problem for some convex functionals of Brownian motion and related processes

Auteur(s):

Code(s) de Classification MSC:

• 44A60 Moment problems
Résumé: This article provides a short and simple proof of the indeterminacy of \ $% A_{t}=\int_{0}^{t}\exp \left( \sigma B_{s}+\nu s\right) ds$, based on refinements of the Krein criteria recently obtained by A.G. Pakes ([15]) and on some elementary probability arguments. In fact, the indeterminacy of $% A_{t}$ is seen as a special case of a more general result about the indeterminacy of exponential functionals of continuous Gaussian processes. We also investigate some variants of this study, in particular when the exponential function is replaced by a wider class of convex functions and when $t$ is replaced by a random time.