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  • 1 Laboratoire d’Analyse et Probabilités, Université d’Évry Val d’Essonne, Boulevard F. Mitterrand, F-91025 Évry Cedex, France
  • 2 Institut Elie Cartan, Université Henri Poincaré, B.P. 239, F-54506 Vandœuvre-lès-Nancy Cedex, France
  • 3 Institut Universitaire de France, Laboratoire de Probabilités et Modèles Aléatoires, Université Paris VI et VII, 4 Place Jussieu — Case 188, F-75252 Paris Cedex 05, France
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Abstract  

Strongly inspired by the result due to Carr-Ewald-Xiao that the arithmetic average of geometric Brownian motion is an increasing process in the convex order, we extend this result to integrals of Lévy processes and Gaussian processes. Our method consists in finding an appropriate sheet associated to the original Lévy or Gaussian process, from which the one-dimensional marginals of the integrals will appear to be those of a martingale, thus proving the increase in the convex order property.