## Abstract

Let μ ∈ P_{2} (R^{d} ), where P_{2} (R^{d} ) denotes the space of square integrable probability measures, and consider a Borel-measurable function ϕ : P_{2} (R^{d} ) → R. In this paper we develop an antithetic Monte Carlo estimator (A-MLMC) for ϕ(μ), which achieves sharp error bound under mild regularity assumptions. The estimator takes as input the empirical laws μ^{N} = _{N}^{1 ΣN}_{i}=_{1} δ_{Xi} , where (a) (X_{i} )^{N}_{i}=_{1} is a sequence of i.i.d. samples from μ or (b) (X_{i} )^{N}_{i}=_{1} is a system of interacting particles (diffusions) corresponding to a McKean-Vlasov stochastic differential equation (McKV-SDE). Each case requires a separate analysis. For a mean-field particle system, we also consider the empirical law induced by its Euler discretisation which gives a fully implementable algorithm. As by-products of our analysis, we establish a dimension-independent rate of uniform strong propagation of chaos, as well as an L^{2} estimate of the antithetic difference for i.i.d. random variables corresponding to general functionals defined on the space of probability measures.

Original language | English |
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Pages (from-to) | 1100-1139 |

Number of pages | 40 |

Journal | Annals of Applied Probability |

Volume | 31 |

Issue number | 3 |

DOIs | |

Publication status | Published - 30 Jun 2021 |

## Keywords

- Antithetic multi-level Monte Carlo estimator
- McKean-Vlasov SDEs
- Propagation of chaos
- Wasserstein calculus