Sharp Rate for the Dual Quantization Problem

Abstract

In this paper we establish the sharp rate of the optimal dual quantization problem. The notion of dual quantization was introduced in Pagès and Wilbertz (SIAM J Numer Anal 50(2):747–780, 2012). Dual quantizers, at least in a Euclidean setting, are based on a Delaunay triangulation, the dual counterpart of the Voronoi tessellation on which “regular” quantization relies. This new approach to quantization shares an intrinsic stationarity property, which makes it very valuable for numerical applications.

We establish in this paper the counterpart for dual quantization of the celebrated Zador theorem, which describes the sharp asymptotics for the quantization error when the quantizer size tends to infinity. On the way we establish an extension of the so-called Pierce Lemma by a random quantization argument. Numerical results confirm our choices.

Appendix: Numerical Results for \(\bar d_{n,2}(X)^2\)

In order to support the heuristic argumentation on the intrinsic and rate optimal growth limitation of the truncation error ${\mathbb P}\big (X\notin C_{n}\big )$ induced by the extended dual quantization error modulus, we consider the two dimensional random variable

$$\displaystyle \begin{aligned} X = \Big(W_T, \sup_{0\leq t \leq T}W_t\Big), \end{aligned}$$

where (W _t)_0≤t≤T is a standard Brownian Motion. This example is motivated by the pricing of path-dependent (exotic) options, where this joint distribution plays an important role.

Using a variant of the CVLQ algorithm (see [12]) adapted for the dual quantization modulus inside C _n and the nearest neighbor mapping outside, we have computed a sequence of optimal grids together with the squared dual quantization error $\bar d_{n,2}(X)^2$ and the truncation error ${\mathbb P}\big (X\notin C_{n}\big )$.

These results are reported in Table 11.1 below.

Table 11.1 Numerical results for the dual quantization X

Furthermore, we see in Fig. 11.1 a log-log plot for the convergence of the two rates $\bar d_{n,2}(X)^2$ and ${\mathbb P}\big (X\notin C_{n}\big )$.

Fig. 11.1

Log-log plot of $\bar d_{n,2}(X)^2$ (distortion error) and ${ \mathbb P} \big (X\notin C_{n} \big )$ (truncation) with respect to the grid size n

The distortion rate $\bar d_{n,2}(X)^2$ shows here an absolute stable convergence rate (least-squares fit of the exponent yields − 1.07192) which is consistent with the theoretical optimal rate of $n^{-\frac {2}{d}}$since d = 2. Moreover, the truncation error ${\mathbb P}\big (X\notin C_{n}\big )$ outperforms also in this case the heuristically derived rate of n ⁻¹ and also outperforms the squared “inside” quantization error, which means that even for such an asymmetric and non-spherical distribution of the Brownian motion and its supremum, a second order rate can be achieved.

This confirms again the motivation of the extended dual quantization error as the correction penalization constraint on growth of the convex hull in order to preserve second order stationarity.