An Unbiased Itô Type Stochastic Representation for Transport PDEs: A Toy Example

Gonçalo dos Reis, Greig Smith

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We propose a stochastic representation for a simple class of transport PDEs based on Itô representations. We detail an algorithm using an estimator stemming for the representation that, unlike regularization by noise estimators, is unbiased. We rely on recent developments on branching diffusions, regime switching processes and their representations of PDEs. There is a loose relation between our technique and regularization by noise, but contrary to the latter, we add a perturbation and immediately its correction. The method is only possible through a judicious choice of the diffusion coefficient σ. A key feature is that our approach does not rely on the smallness of σ, in fact, our σ is strictly bounded from below which is in stark contrast with standard perturbation techniques. This is critical for extending this method to non-toy PDEs which have nonlinear terms in the first derivative where the usual perturbation technique breaks down. The examples presented show the algorithm outperforming alternative approaches. Moreover, the examples point toward a potential algorithm for the fully nonlinear case where the method of characteristics break down.

Original languageEnglish
Title of host publicationFrontiers in Stochastic Analysis - BSDEs, SPDEs and their Applications - Selected, Revised and Extended Contributions
EditorsSamuel N. Cohen, István Gyöngy, Gon?alo dos Reis, David Siska, Lukasz Szpruch
Place of PublicationCham
Number of pages40
ISBN (Electronic)978-3-030-22285-7
ISBN (Print)978-3-030-22284-0
Publication statusPublished - 2019
EventInternational Workshop on BSDEs, SPDEs and their Applications, BSDE-SPDE 2017 - Edinburgh, United Kingdom
Duration: 3 Jul 20177 Jul 2017

Publication series

NameSpringer Proceedings in Mathematics and Statistics
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017


ConferenceInternational Workshop on BSDEs, SPDEs and their Applications, BSDE-SPDE 2017
Country/TerritoryUnited Kingdom


  • Monte Carlo methods
  • Probabilistic methods for PDEs
  • Regime switching diffusion


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