2023/10/19 by Ariel Neufeld, Neufeld, Ariel, Sizhou Wu +1 · 1 citation
Economics, Econometrics and Finance · Engineering · Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Fluid Dynamics and Turbulent Flows #Numerical Analysis (math.NA) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2310.12545
openalex publication_date 2023/10/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce a multilevel Picard approximation algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities whose coefficient functions do not need to be constant. We also provide a full convergence and complexity analysis of our algorithm. To obtain our main results, we consider a particular stochastic fixed-point equation (SFPE) motivated by the Feynman-Kac representation and the Bismut-Elworthy-Li formula. We show that the PDE under consideration has a unique viscosity solution which coincides with the first component of the unique solution of the stochastic fixed-point equation. Moreover, the gradient of the unique viscosity solution of the PDE exists and coincides with the second component of the unique solution of the stochastic fixed-point equation. Furthermore, we also provide a numerical example in up to 300 dimensions to demonstrate the practical applicability of our multilevel Picard algorithm.