Convergence at first and second order of some approximations of stochastic integrals

International audienceWe consider the convergence of the approximation schemes related to Itô's integral and quadratic variation, which have been developed in [13]. First, we prove that the convergence in the a.s. sense exists when the integrand is Hölder continuous and the integrator is a continuous semimartingale. Second, we investigate the second order convergence in the Brownian motion case