Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean

Bencheikh, Oumaima; Jourdain, Benjamin

doi:10.1051/proc/201965219

Cited by 11 publications

(14 citation statements)

References 14 publications

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“…where Φ : P(T d ) → R is a test function chosen within a suitable class. This new direction of research has been introduced in independent works [5,45,53,54]. These works presented novel weak estimates of propagation of chaos for various forms of test functions Φ: Φ is linear in measure in [5], i.e.…”

Section: Propagation Of Chaosmentioning

confidence: 99%

“…This new direction of research has been introduced in independent works [5,45,53,54]. These works presented novel weak estimates of propagation of chaos for various forms of test functions Φ: Φ is linear in measure in [5], i.e. Φ(µ) := T d F (x)µ(dx) for some function F : T d → R; Φ is a polynomial function in [53,54], i.e.…”

Section: Propagation Of Chaosmentioning

confidence: 99%

“…A similar idea has been used in [16] in order to study the convergence problem for mean field games, up to the difference that the equation for the semigroup then becomes a nonlinear equation. Also, another recent work [25] provides an extension of [5,45,54] in the form of a weak error expansion.…”

Section: Propagation Of Chaosmentioning

confidence: 99%

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Uniform in time weak propagation of chaos on the torus

Delarue¹,

Tse²

2021

Preprint

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We address the long time behaviour of weakly interacting diffusive particle systems on the ddimensional torus. Our main result is to show that, under certain regularity conditions, the weak error between the empirical distribution of the particle system and the theoretical law of the limiting process (governed by a McKean-Vlasov stochastic differential equation) is of the order O(1/N ), uniform in time on [0, ∞), where N is the number of particles in the interacting diffusion. This comprises general interaction terms with a small enough mean-field dependence together with interactions terms driven by an H-stable potential. Our approach relies on a systematic analysis of the long-time behaviour of the derivatives of the semigroup generated by the McKean-Vlasov SDE, which may be explicitly computed through the linearised Fokker-Planck equation. Ergodic estimates for the latter hence play a key role in our analysis. We believe that this strategy is flexible enough to cover a wider broad of situations. To wit, we succeed in adapting it to the super-critical Kuramoto model, for which the corresponding McKean-Vlasov equation has several invariant measures. * F. Delarue acknowledges the financial support of the French ANR project ANR-16-CE40-0015-01 on "Mean Field Games" and of the French ANR projet ANR-19-P3IA-0002 "3IA Côte d'Azur -Nice -Interdisciplinary Institute for Artificial Intelligence". † The research of A. Tse benefited from the support of the "Chaire Risques Financiers", Fondation du Risque.

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Section: Propagation Of Chaosmentioning

confidence: 99%

Section: Propagation Of Chaosmentioning

confidence: 99%

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Uniform in time weak propagation of chaos on the torus

Delarue¹,

Tse²

2021

Preprint

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“…An alternative to the empirical measure approximation of (1.2) is to use projection-type estimation of the marginal densities [7] where the error analysis requires differentiability of the coefficients. Variance reduction technique have been analysed for the class of MV-SDE, namely, importance sampling [20], antithetic multilevel Monte Carlo sampling [4] and antithetic sampling [8]. There also recent progress in the jump-diffusion setting [2,10].…”

Section: Introductionmentioning

confidence: 99%

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

Chen,

Reis

2021

Preprint

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We present an implicit Split-Step explicit Euler type Method (dubbed SSM) for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth in space, Lipschitz in measure and non-constant Lipschitz diffusion coefficient. The scheme is designed to leverage the structure induced by the interacting particle approximation system, including parallel implementation and the solvability of the implicit equation.The scheme attains the classical 1/2 root mean square error (rMSE) convergence rate in stepsize and closes the gap left by [18] regarding efficient implicit methods and their convergence rate for this class of McKean-Vlasov SDEs. A sufficient condition for mean-square contractivity of the scheme is presented. Several numerical examples are presented including a comparative analysis to other known algorithms for this class (taming and adaptive time-stepping) across parallel and non-parallel implementations.

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“…has been recently investigated in several papers [19,2,5,6]. Here (W t ) t≥0 is a d-dimensional Brownian motion independent from the initial R n -valued random vector X 0 , W i , Xi 0 i≥1 are i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation law

Bencheikh,

Jourdain

2019

Preprint

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In this paper, we analyse the rate of convergence of a system of N interacting particles with meanfield rank based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by Kolli and Shkolnikhov[18] to check trajectorial propagation of chaos with optimal rate N −1/2 to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by Bossy [3] to check convergence in L 1 (R) with rate O 1 √ N + h of the empirical cumulative distribution function of the Euler discretization with step h of the particle system to the solution of a one dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves in O 1 N + h . We provide numerical results which confirm our theoretical estimates.

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Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean

Cited by 11 publications

References 14 publications

Uniform in time weak propagation of chaos on the torus

Uniform in time weak propagation of chaos on the torus

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation law

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