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Rate of convergence for particle approximation of PDEs in Wasserstein space

Part of: Stochastic analysis Miscellaneous topics - Partial differential equations Special processes

Published online by Cambridge University Press:  28 July 2022

Maximilien Germain ,
Huyên Pham Open the ORCID record for Huyên Pham [Opens in a new window]  and
Xavier Warin
Maximilien Germain*
Affiliation:
EDF R&D, LPSM, and Université Paris Cité
Huyên Pham*
Affiliation:
LPSM, Université Paris Cité, FiME, and CREST ENSAE
Xavier Warin*
Affiliation:
EDF R&D, and FiME
*
*Email address: mgermain@lpsm.paris
**Email address: pham@lpsm.paris
***Email address: xavier.warin@edf.fr
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Abstract

We prove a rate of convergence for the N-particle approximation of a second-order partial differential equation in the space of probability measures, such as the master equation or Bellman equation of the mean-field control problem under common noise. The rate is of order $1/N$ for the pathwise error on the solution v and of order $1/\sqrt{N}$ for the $L^2$ -error on its L-derivative $\partial_\mu v$ . The proof relies on backward stochastic differential equation techniques.

Keywords

Particle approximationWasserstein spacecommon noisemean-field controlmaster equationbackward stochastic differential equations

MSC classification

Primary: 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 992 - 1008
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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