Rate of convergence for particle approximation of PDEs in Wasserstein space *

Maximilien Germain (1, 2, 3) , Huyên Pham (4, 5) , Xavier Warin (1, 5, 3)

1 EDF - EDF
2 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
3 EDF R&D - EDF R&D
4 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
5 FiME Lab - Laboratoire de Finance des Marchés d'Energie

We prove a rate of convergence for the $N$-particle approximation of a second-order partial differential equation in the space of probability measures, like the Master equation or Bellman equation of 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 equations techniques.

Domaines: Finance quantitative [q-fin.CP]; Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Maximilien Germain, Huyên Pham, Xavier Warin. Rate of convergence for particle approximation of PDEs in Wasserstein space *. 2021. ⟨hal-03154021v2⟩