Sharp Rate for the Dual Quantization Problem

Pagès, Gilles; Wilbertz, Benedikt

doi:10.1007/978-3-319-92420-5_11

Cited by 7 publications

(7 citation statements)

References 14 publications

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“…We finally recall the main result established by Pagès and Wilbertz (2018) on the convergence rate of dual quantization for bounded random vectors. Theorem 3.2 (Pierce Lemma for dual quantization).…”

Section: Dual (Delaunay) Quantizationmentioning

confidence: 95%

Quantization and martingale couplings

Jourdain¹,

Pagès²

2022

ALEA

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Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its quadratic optimal) primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension 1 so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to ∞.

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Section: Dual (Delaunay) Quantizationmentioning

confidence: 95%

Quantization and martingale couplings

Jourdain¹,

Pagès²

2022

ALEA

Self Cite

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“…Furthermore, d p,N (X) → 0 as N → +∞. We recall below the main result on convergence rate of dual quantization for bounded random vectors established in [39]. where, for every r > 0, σ r (X) = inf a∈R d X − a r < +∞.…”

Section: A2 Optimal Delaunay (Dual) Quantizationmentioning

confidence: 99%

“…Dual (or Delaunay) quantization introduced by Pagès and Wilbertz [36] and further studied in [37,38,39] gives another way to preserve the convex order in dimension d = 1 (see the remark after Proposition 10 in [37]) when µ n is compactly supported.…”

Section: Introductionmentioning

confidence: 99%

Convex order, quantization and monotone approximations of ARCH models

Jourdain,

Pagès

2019

Preprint

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We are interested in proposing approximations of a sequence of probability measures in the convex order by finitely supported probability measures still in the convex order. We propose to alternate transitions according to a martingale Markov kernel mapping a probability measure in the sequence to the next and dual quantization steps. In the case of ARCH models and in particular of the Euler scheme of a driftless Brownian diffusion, the noise has to be truncated to enable the dual quantization step. We analyze the error between the original ARCH model and its approximation with truncated noise and exhibit conditions under which the latter is dominated by the former in the convex order at the level of sample-paths. Last, we analyse the error of the scheme combining the dual quantization steps with truncation of the noise according to primal quantization.

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“…Dual (or Delaunay) quantization introduced by Pagès and Wilbertz [43] and further studied in [44,45,46] gives another way to preserve the convex order in dimension d = 1 when using the same grid to quantize both probability measures.…”

Section: Introductionmentioning

confidence: 99%

Quantization and martingale couplings

Jourdain¹,

Pagès²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Quantization provides a very natural way to preserve the convex order when approximating two ordered probability measures by two finitely supported ones. Indeed, when the convex order dominating original probability measure is compactly supported, it is smaller than any of its dual quantizations while the dominated original measure is greater than any of its stationary (and therefore any of its optimal) quadratic primal quantization. Moreover, the quantization errors then correspond to martingale couplings between each original probability measure and its quantization. This permits to prove that any martingale coupling between the original probability measures can be approximated by a martingale coupling between their quantizations in Wassertein distance with a rate given by the quantization errors but also in the much finer adapted Wassertein distance. As a consequence, while the stability of (Weak) Martingale Optimal Transport problems with respect to the marginal distributions has only been established in dimension 1 so far, their value function computed numerically for the quantized marginals converges in any dimension to the value for the original probability measures as the numbers of quantization points go to ∞.

show abstract

Sharp Rate for the Dual Quantization Problem

Cited by 7 publications

References 14 publications

Quantization and martingale couplings

Quantization and martingale couplings

Convex order, quantization and monotone approximations of ARCH models

Quantization and martingale couplings

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