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We devise an explicit method to integrate $\alpha$-stable stochasticdifferential equations (SDEs) with non-Lipschitz coefficients. To mitigateagainst numerical instabilities caused by unbounded increments of the L\'evynoise, we use a deterministic map which has the desired SDE as its homogenisedlimit. Moreover, our method naturally overcomes difficulties in expressing theMarcus integral explicitly. We present an example of an SDE with a naturalboundary showing that our method respects the boundary whereas Euler-Maruyamadiscretisation fails to do so. As a by-product we devise an entirelydeterministic method to construct $\alpha$-stable laws.

Simulation of non-Lipschitz stochastic differential equations driven by $\alpha$-stable noise: a method based on deterministic homogenisation