Sharp rate for the dual quantization problem

Sharp asymptotics of the functional quantization problem for Gaussian processes

The sharp asymptotics for the L^2-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.

Intrinsic stationarity for vector quantization: Foundation of dual quantization

We develop a new approach to vector quantization, which guarantees an intrinsic stationarity property that also holds, in contrast to regular quantization, for non-optimal quantization grids. This goal is achieved by replacing the usual nearest neighbor projection operator for Voronoi quantization by a random splitting operator, which maps the random source to the vertices of a triangle of $d$-simplex. In the quadratic Euclidean case, it is shown that these triangles or $d$-simplices make up a Delaunay triangulation of the underlying grid. Furthermore, we prove the existence of an optimal grid…

Itô过程经验路径律的Sharp Wasserstein收敛速率

Sharp functional quantization and empirical Wasserstein rates for Itô processes

该研究针对Itô过程经验路径律，建立了(log N)^{-1/2}的Sharp Wasserstein收敛速率，采用自适应随机时间区间划分方法，其结果可应用于路径依赖SDE及McKean-Vlasov粒子系统相关估计。

Convex order, quantization and monotone approximations of ARCH models

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…

Dual Quantization for random walks with application to credit derivatives

We propose a new Quantization algorithm for the approximation of inhomogeneous random walks, which are the key terms for the valuation of CDO-tranches in latent factor models. This approach is based on a dual quantization operator which posses an intrinsic stationarity and therefore automatically leads to a second order error bound for the weak approximation. We illustrate the numerical performance of our methods in case of the approximation of the conditional tranche function of synthetic CDO products and draw comparisons to the approximations achieved by the saddlepoint method and Stein's me…

Uniform decomposition of probability measures: quantization, classification, rate of convergence

The study of finite approximations of probability measures has a long history. In (Xu and Berger, 2017), the authors focus on constrained finite approximations and, in particular, uniform ones in dimension $d=1$. The present paper gives an elementary construction of a uniform decomposition of probability measures in dimension $d\geq 1$. This decomposition is then used to give upper-bounds on the rate of convergence of the optimal uniform approximation error. These bounds appear to be the generalization of the ones obtained in (Xu and Berger, 2017) and to be sharp for generic probability measur…