Filtering the Feynman--KAC Formula

Abstract

Certain linear partial differential equations involving a Laplacian operator have formulas for their pointwise solutions as functions of Brownian motion. The Feynman--Kac formula, one such, for a heat equation with an additive linear term is smoothed by computing a sequence of conditional expectations adapted to a filtration. Certain additional smoothing of one-dimensional path integrals makes sophisticated numerical rules such as Hermite and Simpson, applied in tandem, effective. The work to get the bias below an error tolerance $\xi$ is $O( \xi^{ - 1/2 \,-\, \epsilon } ) $ with $\epsilon$ an arbitrarily small positive number, whereas it is at least order $\xi^{ - 2} $ for a benchmark method that has been used in practice. By combining the algorithm that yields that bias complexity with randomized quasi-Monte Carlo, the work to get the pointwise root mean square error (RMSE) below $\xi$ is $\, O(\xi^{-7/6 \,-\, \epsilon }) $ for each fixed spatial dimension d. A slightly more intricate algorithm reduces bias complexity---leading to $O(\xi^{-20/21 \,-\, \epsilon}) $ RMSE complexity. The latter is lower than an order $\xi^{-(d+1)/2}$ sharp lower bound on complexity for conventional solvers for each fixed d, the difference increasing rapidly with d---assuming that the number of points where a solution is desired is O(1) for each d.