Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

TL;DR

This paper introduces a novel branching diffusion representation for semilinear parabolic PDEs with polynomial nonlinearities, enabling efficient Monte Carlo methods for high-dimensional problems with error estimates.

Contribution

It extends classical representations to include gradient nonlinearities using automatic differentiation, facilitating Monte Carlo simulation for high-dimensional PDEs.

Findings

First numerical method for high-dimensional nonlinear PDEs with error bounds

Representation suitable for Monte Carlo simulation

Demonstrated efficiency through high-dimensional numerical experiments

Abstract

We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod (1964), Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in the pair $(u, Du)$, where $u$ is the solution of the PDE with space gradient $Du$. Similar to the previous literature, our result requires a non-explosion condition which restrict to "small maturity" or "small nonlinearity" of the PDE. Our main ingredient is the automatic differentiation technique as in Henry Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts, which allows to account for the nonlinearities in the gradient. As a consequence, the particles of our branching diffusion are marked by the nature of the nonlinearity. This new representation has very important numerical implications as it is suitable for Monte Carlo simulation. Indeed, this provides the first numerical method for high dimensional nonlinear PDEs with error estimate induced by the dimension-free Central limit theorem. The complexity is also easily seen to be of the order of the squared dimension. The final section of this paper illustrates the efficiency of the algorithm by some high dimensional numerical experiments.