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.