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

Summary This thesis contains two parts. In the first part, we prove two limit theorems of optimal quantization. The first limit theorem is the characterization of the convergence under the Wasserstein distance of a sequence of probability measures by the simple convergence of the quantization error functions. These results are established in Rd and also in a separable Hilbert space. The second limit theorem shows the speed of convergence of the optimal grids and the quantization performance for a sequence of probability measures which converge under the Wasserstein distance, in particular the empirical measure. The second part of this thesis focuses on the approximation and simulation of the McKean-Vlasov equation. We start this part by proving, by Feyel's method (see Bouleau (1988) [Section 7]), 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 condition that the coefficient functions b and σ are lipschitzian. Then, the convergence speed of the theoretical Euler scheme of the McKean-Vlasov equation is established and also the convex order functional results for the McKean-Vlasov equations with b(t,x,μ) = αx+β, α,β ∈ R. In the last chapter, the error of the particle method, several quantization-based schemes and a hybrid particle-quantization scheme are analyzed. At the end, two example simulations are illustrated: the Burgers equation (Bossy and Talay (1997)) in dimension 1 and the FitzHugh-Nagumo neural network (Baladron et al. (2012)) in dimension 3.