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Optimal Quantization : Limit Theorem, Clustering and Simulation of the McKean-Vlasov Equation

Abstract : This thesis contains two parts. The first part addresses two limit theorems related to optimal quantization. The first limit theorem is the characterization of the convergence in the Wasserstein distance of probability measures by the pointwise convergence of Lp-quantization error functions on Rd and on a separable Hilbert space. The second limit theorem is the convergence rate of the optimal quantizer and the clustering performance for a probability measure sequence (μn)n∈N∗ on Rd converging in the Wasserstein distance, especially when (μn)n∈N∗ are the empirical measures with finite second moment but possibly unbounded support. The second part of this manuscript is devoted to the approximation and the simulation of the McKean-Vlasov equation, including several quantization based schemes and a hybrid particle-quantization scheme. We first give a proof of the existence and uniqueness of a strong solution of the McKean- Vlasov equation dXt = b(t, Xt, μt)dt + σ(t, Xt, μt)dBt under the Lipschitz coefficient condition by using Feyel’s method (see Bouleau (1988)[Section 7]). Then, we establish the convergence rate of the “theoretical” Euler scheme and as an application, we establish functional convex order results for scaled McKean-Vlasov equations with an affine drift. In the last chapter, we prove the convergence rate of the particle method, several quantization based schemes and the hybrid scheme. Finally, we simulate two examples: the Burger’s equation (Bossy and Talay (1997)) in one dimensional setting and the Network of FitzHugh-Nagumo neurons (Baladron et al. (2012)) in dimension 3.
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Submitted on : Wednesday, September 30, 2020 - 6:32:06 PM
Last modification on : Friday, August 5, 2022 - 3:00:08 PM
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  • HAL Id : tel-02954146, version 2


Yating Liu. Optimal Quantization : Limit Theorem, Clustering and Simulation of the McKean-Vlasov Equation. Probability [math.PR]. Sorbonne Université, 2019. English. ⟨NNT : 2019SORUS215⟩. ⟨tel-02954146v2⟩



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