Optimal quadratic quantization for numerics: the Gaussian case

Gilles Pagès 1  and Jacques Printems 2
  • 1 Laboratoire de Probabilités et Modèles Aléatoires, CNRS UMR 7599, Université Paris 6, case 188, 4, pl. Jussieu, F-75252 Paris Cedex 5. E-mail: gpa@ccr.jussieu.fr
  • 2 INRIA, MathFi project and Centre de Mathématiques, CNRS UMR 8050, Université Paris 12, 61, av. du Général de Gaulle, F-94010 Créteil. E-mail: printems@univ-paris12.fr

Optimal quantization has been recently revisited in multi-dimensional numerical integration, multi-asset American option pricing, control theory and nonlinear filtering theory. In this paper, we enlighten some numerical procedures in order to get some accurate optimal quadratic quantization of the Gaussian distribution in one and higher dimensions. We study in particular Newton method in the deterministic case (dimension d = 1) and stochastic gradient in higher dimensional case (d ≥ 2). Some heuristics are provided which concern the step in the stochastic gradient method. Finally numerical examples borrowed from mathematical finance are used to test the accuracy of our Gaussian optimal quantizers.

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This quarterly published journal presents original articles on the theory and applications of Monte Carlo and Quasi-Monte Carlo methods. Launched in 1995 the journal covers all stochastic numerics topics with emphasis on the theory of Monte Carlo methods and new applications in all branches of science and technology.

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