In this paper we introduce a multilevel Picard approximation algorithm for semilinear parabolic partial integro-differential equations (PIDEs). We prove that the numerical approximation scheme converges to the unique viscosity solution of the PIDE under consideration. To that end, we derive a Feynman-Kac representation for the unique viscosity solution of the semilinear PIDE, extending the classical Feynman-Kac representation for linear PIDEs. Furthermore, we show that the algorithm does not suffer from the curse of dimensionality, i.e. the computational complexity of the algorithm is bounded polynomially in the dimension $d$ and the reciprocal of the prescribed accuracy $\varepsilon$. We also provide a numerical example in up to 10'000 dimensions to demonstrate its applicability.

翻译：本文提出了一种针对半线性抛物型偏积分微分方程（PIDEs）的多层Picard逼近算法。我们证明了该数值逼近格式收敛于所考虑PIDE的唯一粘性解。为此，我们推导了半线性PIDE唯一粘性解的Feynman-Kac表示，将线性PIDEs的经典Feynman-Kac表示进行了推广。此外，我们证明该算法不受维数灾难的影响，即算法的计算复杂度在维度$d$和预设精度$\varepsilon$的倒数上呈多项式有界。我们还提供了高达10'000维的数值示例，以展示其适用性。