Limit theorems for compensated weighted sums and application to numerical approximations

In this paper, we consider a "compensated" random sum that arises from numerical approximation of stochastic integrations and differential equations. We show that the compensated sum exhibits some surprising cancellations among its components, a property which allows to transform it into a Skorohod-type Riemann sum. We then establish limit theorem for the compensated sum based on study of the Skorohod-type Riemann sum. Our proof employs techniques from Malliavin calculus and rough path. We apply our limit theorem result to the Euler approximation method for stochastic integrals and additive stochastic differential equations, filling a notable gap in this area of research. We show that the Euler method converges to the solution at the rate $(1/n)^{H+1/2}$, and that this rate is exact in the sense that the asymptotic error distribution solves a linear differential equation.