Solving Elliptic Equations with Brownian Motion: Bias Reduction and Temporal Difference Learning

Cameron Martin, Hongyuan Zhang, Julia Costacurta, Mihai Nica and Adam R Stinchcombe ()
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Cameron Martin: University of Toronto
Hongyuan Zhang: Carnegie Mellon University
Julia Costacurta: Stanford University
Mihai Nica: University of Guelph
Adam R Stinchcombe: University of Toronto

Methodology and Computing in Applied Probability, 2022, vol. 24, issue 3, 1603-1626

Abstract: Abstract The Feynman-Kac formula provides a way to understand solutions to elliptic partial differential equations in terms of expectations of continuous time Markov processes. This connection allows for the creation of numerical schemes for solutions based on samples of these Markov processes which have advantages over traditional numerical methods in some cases. However, naïve numerical implementations suffer from issues related to statistical bias and sampling efficiency. We present methods to discretize the stochastic process appearing in the Feynman-Kac formula that reduce the bias of the numerical scheme. We also propose using temporal difference learning to assemble information from random samples in a way that is more efficient than the traditional Monte Carlo method.

Keywords: Feynman-Kac formula; Monte Carlo; Temporal difference learning; Brownian motion; Elliptic equation; 65N75; 65C05 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11009-021-09871-9

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