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Semilinear, $N-$dimensional stochastic differential equations (SDEs) drivenby additive L\'evy noise are investigated. Specifically, given$\alpha\in\left(\frac{1}{2},1\right)$, the interest is on SDEs driven by$2\alpha-$stable, rotation-invariant processes obtained by subordination of aBrownian motion. An original connection between the time-dependent Markovtransition semigroup associated with their solutions and Kolmogorov backwardequations in mild integral form is established via regularization-by-noisetechniques. Such a link is the starting point for an iterative method whichallows to approximate probabilities related to the SDEs with a single batch ofMonte Carlo simulations as several parameters change, bringing a compellingcomputational advantage over the standard Monte Carlo approach. This methodalso pertains to the numerical computation of solutions to high-dimensionalintegro-differential Kolmogorov backward equations. The scheme, and inparticular the first order approximation it provides, is then applied for twononlinear vector fields and shown to offer satisfactory results in dimension$N=100$.

Probability computation for high-dimensional semilinear SDEs driven by isotropic $\alpha-$stable processes via mild Kolmogorov equations