Matching the test function's spatial regularity to that of the singular drift (Lebesgue, Hölder or negative Besov) improves or preserves weak convergence rates for discretization of SDEs with stable noise.
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Weak error for SDEs with additive stable noise and singular drift: choose the test function in the same space as the drift!
Matching the test function's spatial regularity to that of the singular drift (Lebesgue, Hölder or negative Besov) improves or preserves weak convergence rates for discretization of SDEs with stable noise.