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A backward Monte-Carlo method for solving parabolic partial differential equations

Johan Carlsson

TL;DR

The paper addresses solving linear parabolic PDEs with Monte-Carlo methods while mitigating statistical noise, especially in tail regions. It introduces a backward Monte-Carlo scheme derived from the Feynman-Kac representation, tracing trajectories backward in time and evaluating the solution as the expectation of the initial condition at t=0. The key contributions are a delta-free, smooth solution with roughly constant relative error across phase space, and the ability to focus sampling where it matters, which aids efficiency in high-dimensional or localized problems. The method is compatible with existing MC codes and extends naturally to general linear parabolic equations via the associated stochastic differential equations.

Abstract

A new Monte-Carlo method for solving linear parabolic partial differential equations is presented. Since, in this new scheme, the particles are followed backward in time, it provides great flexibility in choosing critical points in phase-space at which to concentrate the launching of particles and thereby minimizing the statistical noise of the sought solution. The trajectory of a particle, Xi(t), is given by the numerical solution to the stochastic differential equation naturally associated with the parabolic equation. The weight of a particle is given by the initial condition of the parabolic equation at the point Xi(0). Another unique advantage of this new Monte-Carlo method is that it produces a smooth solution, i.e. without delta-functions, by summing up the weights according to the Feynman-Kac formula.

A backward Monte-Carlo method for solving parabolic partial differential equations

TL;DR

The paper addresses solving linear parabolic PDEs with Monte-Carlo methods while mitigating statistical noise, especially in tail regions. It introduces a backward Monte-Carlo scheme derived from the Feynman-Kac representation, tracing trajectories backward in time and evaluating the solution as the expectation of the initial condition at t=0. The key contributions are a delta-free, smooth solution with roughly constant relative error across phase space, and the ability to focus sampling where it matters, which aids efficiency in high-dimensional or localized problems. The method is compatible with existing MC codes and extends naturally to general linear parabolic equations via the associated stochastic differential equations.

Abstract

A new Monte-Carlo method for solving linear parabolic partial differential equations is presented. Since, in this new scheme, the particles are followed backward in time, it provides great flexibility in choosing critical points in phase-space at which to concentrate the launching of particles and thereby minimizing the statistical noise of the sought solution. The trajectory of a particle, Xi(t), is given by the numerical solution to the stochastic differential equation naturally associated with the parabolic equation. The weight of a particle is given by the initial condition of the parabolic equation at the point Xi(0). Another unique advantage of this new Monte-Carlo method is that it produces a smooth solution, i.e. without delta-functions, by summing up the weights according to the Feynman-Kac formula.

Paper Structure

This paper contains 8 sections, 28 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Feynman-Kac formula
  3. Backward Monte-Carlo method
  4. Discussion
  5. Summary
  6. Stochastic calculus and Itô's formula
  7. General linear parabolic PDE
  8. Langevin equation

Figures (2)

  • Figure 1: The backward Monte-Carlo method. The trajectories as given by the backward Monte-Carlo difference equation of motion (\ref{['eq:motion']}).
  • Figure 2: Conventional, forward Monte-Carlo method with the same weights as the backward method. The trajectories as given by the forward Monte-Carlo difference equation of motion (\ref{['eq:motion-forward']}).