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High order weak approximation schemes for Lévy-driven SDEs

We propose new jump-adapted weak approximation schemes for stochastic differential equations driven by pure-jump Lévy processes. The idea is to replace the driving Lévy process $Z$ with a finite intensity process which has the same Lévy measure outside a neighborhood of zero and matches a given number of moments of $Z$. By matching 3 moments we construct a scheme which works for all Lévy measures and is superior to the existing approaches both in terms of convergence rates and easiness of implementation. In the case of Lévy processes with stable-like behavior of small jumps, we construct schemes with arbitrarily high rates of convergence by matching a sufficiently large number of moments.