Optimal sampling design for approximation of stochastic Itô integrals with application to the nonlinear Lebesgue integration

We study the approximation of stochastic integrals in the Itô sense. We establish the exact convergence rate of the minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Brownian motion. We provide a construction of optimal schemes which asymptotically attain established minimal errors. Using results obtained for the Itô integration we investigate the minimal asymptotic errors for the problem of nonlinear Lebesgue integration in the average case setting.