Multi-dimensional Avikainen's estimates
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Avikainen proved the estimate E[|f(X)-f(X)|^q] ≤ C(p,q) E[|X-X|^p]^1/p+1 for p,q ∈ [1,∞), one-dimensional random variables X with the bounded density function and X, and a function f of bounded variation in R. In this article, we will provide multi-dimensional analogues of this estimate for functions of bounded variation in R^d, Orlicz-Sobolev spaces, Sobolev spaces with variable exponents and fractional Sobolev spaces. The main idea of our arguments is to use Hardy-Littlewood maximal estimates and pointwise characterizations of these function spaces. We will apply main statements to numerical analysis on irregular functionals of a solution to stochastic differential equations based on the Euler-Maruyama scheme and the multilevel Monte Carlo method, and to estimates of the L^2-time regularity of decoupled forward-backward stochastic differential equations with irregular terminal conditions.
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