We present an explicit numerical approximation scheme for the effective
simulation of solutions to a multivariate stochastic differential
equation (SDE) with a superlinearly growing dissipative drift, driven
by a multiplicative heavy-tailed Lévy process. The scheme combines the
well-known Euler method with a Lie-Trotter-type splitting technique.
The specific ordering of the splitting terms enables the approximation
to capture all finite moments of the true solution. In the special case
of SDEs driven solely by Brownian motion, our numerical scheme
preserves the solution's superexponential moments. We prove strong
convergence of approximations and determine the order of convergence.