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Several finite difference methods are proposed for the infinitesimalgenerator of 1D asymmetric $\alpha$-stable L\'{e}vy motions, based on the factthat the operator becomes a multiplier in the spectral space. These methodstake the general form of a discrete convolution, and the coefficients (or theweights) in the convolution are chosen to approximate the exact multiplierafter appropriate transform. The accuracy and the associatedadvantages/disadvantages are also discussed, providing some guidance on thechoice of the right scheme for practical problems, like in the calculation ofmean exit time for random processes governed by general asymmetric$\alpha$-stable motions.

Finite Difference Methods for the generator of 1D asymmetric alpha-stable L\'{e}vy motions