Multilevel Picard approximation algorithm for semilinear partial integro-differential equations and its complexity analysis

TL;DR

The paper develops a multilevel Picard (MLP) framework for semilinear PIDEs with nonlocal (jump) terms, establishing a Feynman–Kac representation and proving convergence to the unique viscosity solution while overcoming the curse of dimensionality. It integrates stochastic fixed-point theory, mollification, and jump-augmented SDEs to derive contraction properties and rigorous error/complexity bounds, showing polynomial dependence on the dimension $d$ and the inverse accuracy $1/oldsymbol{initeepsilon}$. A key strength is the inclusion of a nonlocal Lévy term and a concrete numerical example in very high dimensions (up to $10^4$), demonstrating practical applicability. The work offers both Euler-based and non-Euler MLP variants, with detailed convergence, stability, and complexity analyses that advance high-dimensional nonlinear PIDEs in finance and physics.

Abstract

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.

Theorems & Definitions (57)

• Remark 2.4
• Definition 2.7
• Remark 2.8
• Theorem 2.9
• Theorem 2.10
• Proposition 3.1
• proof
• Lemma 3.2
• proof
• Lemma 4.2