Multilevel Picard algorithm for general semilinear parabolic PDEs with gradient-dependent nonlinearities

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, if the PDE admits a strong solution, then the gradient of the unique solution of the PDE coincides with the second component of the unique solution of the stochastic fixed-point equation.