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

## TL;DR

This paper develops finite difference methods to approximate the generator of 1D asymmetric alpha-stable Levy motions, facilitating numerical analysis of related stochastic processes.

## Contribution

It introduces spectral-based finite difference schemes with specific convolution weights for the generator of asymmetric alpha-stable Levy motions, analyzing their accuracy and practical applicability.

## Key findings

- Methods effectively approximate the generator in spectral space.
- Different schemes have distinct accuracy and computational advantages.
- Guidance provided for selecting suitable schemes for practical problems.

## Abstract

Several finite difference methods are proposed for the infinitesimal generator of 1D asymmetric $\alpha$-stable L\'{e}vy motions, based on the fact that the operator becomes a multiplier in the spectral space. These methods take the general form of a discrete convolution, and the coefficients (or the weights) in the convolution are chosen to approximate the exact multiplier after appropriate transform. The accuracy and the associated advantages/disadvantages are also discussed, providing some guidance on the choice of the right scheme for practical problems, like in the calculation of mean exit time for random processes governed by general asymmetric $\alpha$-stable motions.