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 prescribed reciprocal of the accuracy $\varepsilon$.

翻译：在本文中,我们引入了半线性抛物线部分异种方程的多级Picard近似算法(PIDEs ) 。 我们证明数字近似法与所考虑的PIDE的独特粘度解决方案相融合。 为此,我们为半线性PIDE的独特粘度解决方案制作了Feynman-Kac代表法,将古典Feynman-Kac代表法延伸至线性 PIDE。 此外,我们证明该算法没有受到维度诅咒的影响,也就是说,该算法的计算复杂性在维度上是多维维的,而规定的准确度是$\varepsilon$的对等。