Asymptotically optimal quantization schemes for Gaussian processes on Hilbert spaces

Luschgy, Harald; Pagès, Gilles; Wilbertz, Benedikt

doi:10.1051/ps:2008026

Cited by 16 publications

(16 citation statements)

References 19 publications

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“…≈ λ 1 /10. This is probably one reason for which former attempts to produce good quantization of the Brownian motion first focused on other kinds of quantizers like scalar product quantizers (see [44] and Section 7.4 below) or d-dimensional block product quantizations (see [56] and [35]).…”

Section: Numerical Optimization Of Quadratic Functional Quantizationmentioning

confidence: 99%

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Quadratic Optimal Functional Quantization of Stochastic Processes and Numerical Applications

Pagès¹

2008

Monte Carlo and Quasi-Monte Carlo Methods 2006

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In this paper, we present an overview of the recent developments of functional quantization of stochastic processes, with an emphasis on the quadratic case. Functional quantization is a way to approximate a process, viewed as a Hilbertvalued random variable, using a nearest neighbour projection on a finite codebook. A special emphasis is made on the computational aspects and the numerical applications, in particular the pricing of some path-dependent European options. IntroductionFunctional quantization is a way to discretize the path space of a stochastic process. It has been extensively investigated since the early 2000's by several authors (see among others [29], [31], [12], [9], [30], etc). It first appeared as a natural extension of the Optimal Vector Quantization theory of (finitedimensional) random vectors which finds its origin in the early 1950's for signal processing (see [15] or [17]).Let us consider a Hilbertian setting. One considers a random vector X defined on a probability space (Ω, A, P) taking its values in a separable Hilbert space (H, (.|.) H ) (equipped with its natural Borel σ-algebra) and satisfying E|X| 2 < +∞. When H is an Euclidean space (R d ), one speaks about Vector Quantization. When H is an infinite dimensional space like L 2 T := L 2 ([0, T ], dt) (endowed with the usual Hilbertian norm |f | L 2 T := (T 0 f 2 (t)dt) 1 2 ) one speaks of functional quantization (denoted L 2 T from now on). A (bi-measurable) stochastic process (X t ) t∈[0,T ] defined on (Ω, A, P) satisfying |X(ω)| L 2 T < +∞ P(dω)-a.s. can always be seen, once possibly modified on a P-negligible set, as an L 2 T -valued random variable. Although we will focus on the Hilbertian framework, other choices are possible for H, in particular some more general Banach settings like L p ([0, T ], dt) or C([0, T ], R) spaces.This paper is organized as follows: in Sections 2 we introduce quadratic quantization in a Hilbertian setting. In Section 3, we focus on optimal quantization, including some extensions to non quadratic quantization. Section 4 is

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Section: Numerical Optimization Of Quadratic Functional Quantizationmentioning

confidence: 99%

“…• d-dimensional block quantization is also possible, possibly with varying block size, providing a constructive approach to sharp rate, see [56] and [35].…”

Section: Quantization Rate By Product Quantizersmentioning

confidence: 99%

Quadratic Optimal Functional Quantization of Stochastic Processes and Numerical Applications

Pagès¹

2008

Monte Carlo and Quasi-Monte Carlo Methods 2006

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“…A conjecture supported by numerical evidences is that d W (N ) ∼ log N . Recently a first step to this conjecture has been established in [23] by showing that lim inf…”

Section: Optimal Quantization (D = 1)mentioning

confidence: 99%

Convergence of Multi-Dimensional Quantized SDE’s

Pagès

2010

Séminaire De Probabilités XLIII

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We quantize a multidimensional SDE (in the Stratonovich sense) by solving the related system of ODE's in which the d-dimensional Brownian motion has been replaced by the components of functional stationary quantizers. We make a connection with rough path theory to show that the solutions of the quantized solutions of the ODE converge toward the solution of the SDE. On our way to this result we provide convergence rates of optimal quantizations toward the Brownian motion for 1 q -Hölder distance, q > 2, in L p (P).

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“…Under the (H) hypothesis, this suggests to define the partial quantization of S from a partial quantization ‹ X I,Γ of X by replacing X by ‹ X I,Γ in the SDE (22). We define the partial quantization S I,Γ as the process whose conditional distribution given that Y falls in the Voronoi cell of γ k is the strong solution of the same SDE where X is replaced by the K-L generalized bridge with end-point γ k .…”

Section: Partial Functional Quantization Of Stochastic Differential Ementioning

confidence: 99%

Partial functional quantization and generalized bridges

Corlay¹

2014

Bernoulli

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In this article, we develop a new approach to functional quantization, which consists in discretizing only a finite subset of the Karhunen-Loève coordinates of a continuous Gaussian semimartingale X.Using filtration enlargement techniques, we prove that the conditional distribution of X knowing its first Karhunen-Loève coordinates is a Gaussian semimartingale with respect to a bigger filtration. This allows us to define the partial quantization of a solution of a stochastic differential equation with respect to X by simply plugging the partial functional quantization of X in the SDE.Then we provide an upper bound of the L p -partial quantization error for the solution of SDEs involving the L p+ε -partial quantization error for X, for ε > 0. The a.s. convergence is also investigated.Incidentally, we show that the conditional distribution of a Gaussian semimartingale X, knowing that it stands in some given Voronoi cell of its functional quantization, is a (non-Gaussian) semimartingale. As a consequence, the functional stratification method developed in [6] amounted, in the case of solutions of SDEs, to using the Euler scheme of these SDEs in each Voronoi cell. (1) is an optimal quantizer of X. The corresponding quantization error is denoted by A solution toOne usually drops the p subscript in the quadratic case (p = 2). This problem, initially investigated as a signal discretization method [10], has then been introduced in numerical probability to devise cubature methods [24] or to solve multidimensional stochastic control problems [3]. Since the early 2000's, the infinite-dimensional setting has been extensively investigated from both constructive numerical and theoretical viewpoints with a special attention paid to functional quantization, especially in the quadratic case [18] but also in some other Banach spaces [29]. Stochastic processes are viewed as random variables taking values in functional spaces.

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Asymptotically optimal quantization schemes for Gaussian processes on Hilbert spaces

Cited by 16 publications

References 19 publications

Quadratic Optimal Functional Quantization of Stochastic Processes and Numerical Applications

Quadratic Optimal Functional Quantization of Stochastic Processes and Numerical Applications

Convergence of Multi-Dimensional Quantized SDE’s

Partial functional quantization and generalized bridges

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