Unbiased local solutions of partial differential equations via the Feynman-Kac Identities

TL;DR

This paper introduces methods to obtain unbiased local solutions of PDEs using the Feynman-Kac formula by combining exact diffusion path simulation and debiasing techniques, improving statistical inference accuracy.

Contribution

It demonstrates how recent advances in exact SDE simulation and debiasing enable unbiased Monte Carlo solutions for PDEs via the FKF, overcoming previous discretization biases.

Findings

Unbiased solutions for a wide range of PDE models are achievable.

Exact diffusion path simulation eliminates discretization bias.

Debiasing methods enable unbiased estimates from biased sequences.

Abstract

The Feynman-Kac formulae (FKF) express local solutions of partial differential equations (PDEs) as expectations with respect to some complementary stochastic differential equation (SDE). Repeatedly sampling paths from the complementary SDE enables the construction of Monte Carlo estimates of local solutions, which are more naturally suited to statistical inference than the numerical approximations obtained via finite difference and finite element methods. Until recently, simulating from the complementary SDE would have required the use of a discrete-time approximation, leading to biased estimates. In this paper we utilize recent developments in two areas to demonstrate that it is now possible to obtain unbiased solutions for a wide range of PDE models via the FKF. The first is the development of algorithms that simulate diffusion paths exactly (without discretization error), and so make it possible to obtain Monte Carlo estimates of the FKF directly. The second is the development of debiasing methods for SDEs, enabling the construction of unbiased estimates from a sequence of biased estimates.