Abstract
This chapter is a first introduction to optimal vector quantization and its application to numerical probability. Optimal quantization produces the best approximation of probability distribution by finitely supported distributions in the sues of the Wasserstein distance. It naturally yields cubature formulas to compute expectations \(\mathbb {E}F(X)\) where X is an \(\mathbb {R}^{d}\)-valued random vector and F a Lipschitz or has a Lipschitz gradient (this applies too to conditional expectations). Thanks to A quantization based Richardson–Romberg extrapolations, tens e cubature formulas are shown to be competitive with regular Monte Carlo simulation at least up to 5 dimensions. A first approach to the computation of (quadratic) optimal quantizers of a given distribution is developed. Quantization is also investigated in Chap. 6 from an algorithmic view point and 11 as a numerical method to price Bermuda and American options.
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Notes
- 1.
The “Gaussian” solid line follows the shape of \(\xi \mapsto \varphi _{\frac{1}{3}}(\xi )= \frac{e^{-\frac{\xi ^2}{2 \cdot 3}}}{\sqrt{3}\sqrt{2\pi }}\), i.e. \({\mathbb P}\big (X\!\in C_i(x^{(N)})\big )\simeq \varphi _{\frac{1}{3}}(x^{(N)}_i)\), \(i=1,\ldots , N\), in a sense to be made precise. This is another property of optimal quantizers which is beyond the scope of this textbook, see e.g. [132].
- 2.
This means that there is a Borel set \(A_{\nu }\!\in \mathcal{B}or({\mathbb R}^d)\) such that \(\lambda _d(A_{\nu })=0\) and \(\nu (A_{\nu })= \nu ({\mathbb R}^d)\). Such a decomposition always exists and is unique.
- 3.
If \(\mu \) and \(\nu \) are distributions on \(\big ({\mathbb R}^d,\mathcal{B}or({\mathbb R}^d)\big )\) with finite pth moment (\(1\le p<+\infty \)), the \(L^p\)-Wasserstein distance between \(\mu \) and \(\nu \) is defined by
$$ \mathcal{W}_1(\mu ,\nu ) = \inf \left\{ \left[ \int _{{\mathbb R}^d\times {\mathbb R}^d} |x-y |^p\,m(dx, dy)\right] ^{\frac{1}{p}}\!, \, m \text { Borel distribution on } {\mathbb R}^d\times {\mathbb R}^d, m(dx\times {\mathbb R}^d)=\mu ,\, m({\mathbb R}^d\times dy)=\nu \right\} . $$When \(p=1\), the Monge–Kantorovich characterization of \(\mathcal {W}_1\) reads as follows:
$$ \mathcal{W}_1(\mu ,\nu ) = \sup \Big \{\Big |\int _{{\mathbb R}^d} f d\mu -\int _{{\mathbb R}^d} f d\nu \Big |,\; f:{\mathbb R}^d\rightarrow {\mathbb R},\;[f]_\mathrm{Lip}\le 1\Big \}. $$Note that the absolute values can be removed in the above characterization of \(\mathcal{W}_1(\mu ,\nu )\) since f and \(-f\) are simultaneously Lipschitz continuous with the same Lipschitz coefficient.
- 4.
A more precise approximation is \(C= 0.24852267852801818 \pm 2.033\ 10^{-7}\) obtained by implementing an ML2R estimator with a target RMSE \(\varepsilon = 3.0\, 10^{-7}\), see Chap. 9.
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Pagès, G. (2018). Optimal Quantization Methods I: Cubatures. In: Numerical Probability. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-90276-0_5
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DOI: https://doi.org/10.1007/978-3-319-90276-0_5
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