A generalized Avikainen's estimate and its applications
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Avikainen provided a sharp upper bound of the difference E[|g(X)-g(X)|^q] by the moments of |X-X| for any one-dimensional random variables X with bounded density and X, and function of bounded variation g. In this article, we generalize this estimate to any one-dimensional random variable X with Hölder continuous distribution function. As applications, we provide the rate of convergence for numerical schemes for solutions of one-dimensional stochastic differential equations (SDEs) driven by Brownian motion and symmetric α-stable with α∈ (1,2), fractional Brownian motion with drift and Hurst parameter H ∈ (0,1/2), and stochastic heat equations (SHEs) with Dirichlet boundary conditions driven by space–time white noise, with irregular coefficients. We also consider a numerical scheme for maximum and integral type functionals of SDEs driven by Brownian motion with irregular coefficients and payoffs which are related to multilevel Monte Carlo method.
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