@article {89,
	title = {Optimal Delaunay and Voronoi quantization schemes for pricing American style options.},
	year = {2011},
	abstract = {We review in this article pure quantization methods for the pricing of multiple exercise options. These quantization methods have the common advantage, that they allow a straightforward implementation of the Backward Dynamic Programming Principle for optimal stopping and stochastic control problems. Moreover we present here for the first time a unified discussion of this topic for Voronoi and Delaunay quantization and illustrate the performances of both methods by several numerical examples.},
	keywords = {American option, Delaunay triangulation, optimal vector quantization, quantization tree, Voronoi diagram},
	author = {Gilles Pag{\`e}s and Benedikt Wilbertz}
}
@article {91,
	title = {Optimal quantization applied to sliced inverse regression},
	year = {2011},
	abstract = {In this paper we consider a semiparametric regression model involving a $d$-dimensional quantitative explanatory variable $X$ and including a dimension reduction of $X$ via an index $\beta{\textquoteright}X$. In this model, the main goal is to estimate the Euclidean parameter and to predict the real response variable $Y$ conditionally to $X$. Our approach is based on sliced inverse regression (SIR) method and optimal quantization in $L^p$-norm. We obtain the convergence of the proposed estimators of  and of the conditional distribution. Simulation studies show the good numerical behavior of the proposed estimators for finite sample size.},
	keywords = {optimal quantization, reduction dimension, semiparametric regression model, sliced inverse regression (SIR)},
	author = {Romain Aza{\"\i}s and Anne G{\'e}gout-Petit and J{\'e}r{\^o}me Saracco}
}
@article {87,
	title = {Partial functional quantization and generalized bridges},
	year = {2011},
	abstract = {In this article, we develop a new approach to functional quantization, which consists in discretizing only the first Karhunen-Lo{\`e}ve coordinates of a continuous Gaussian semimartingale $X$. Using filtration enlargement techniques, we prove that the conditional distribution of $X$ knowing its first Karhunen-Lo{\`e}ve coordinates is a Gaussian semimartingale with respect to its natural filtration. 
This allows 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 SDE involving the $L^{p+\varepsilon}$-partial quantization error for $X$, for $\varepsilon >0$. 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 [bib]CorlayPagesStratification[/bib], amounted in the case of solutions of SDE to use the Euler scheme of these SDE in each Voronoi cell.},
	keywords = {Brownian bridge, Brownian motion, filtration enlargement, functional quantization, Gaussian process, Gaussian semimartingale, Karhunen-Lo{\`e}ve, Ornstein-Uhlenbeck, stratification, vector quantization},
	attachments = {http://quantif.maths-fi.com/sites/default/files/partial quantization.pdf},
	author = {Sylvain Corlay}
}
@article {45,
	title = {Asymptotically optimal quantization schemes for Gaussian processes},
	journal = {ESAIM: PS},
	volume = {14},
	year = {2010},
	pages = {93 - 116},
	abstract = {We describe quantization designs which lead to asymptotically and order optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role to achieve these rates. For the development of a constructive quantization scheme we rely on the knowledge of the eigenvectors of the covariance operator in order to transform the problem into a finite dimensional quantization problem of normal distributions. Furthermore we derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes.},
	keywords = {Brownian motion, functional quantization, Gaussian process, optimal quantizer, Riemann-Liouville process},
	author = {Harald Luschgy and Gilles Pag{\`e}s and Benedikt Wilbertz}
}
@article {85,
	title = {A constructive sharp approach to functional quantization of stochastic processes},
	journal = {Journal of Applied Mathematics},
	volume = {2010},
	year = {2010},
	abstract = {We present a constructive approach to the functional quantization problem of stochastic processes, with an emphasis on Gaussian processes. The approach is constructive, since we reduce the infinite-dimensional functional quantization problem to a finite-dimensional quantization problem that can be solved numerically. Our approach achieves the sharp rate of the minimal quantization error and can be used to quantize the path space for Gaussian processes and also, for example, L{\'e}vy processes.},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Constructive functional quantization.pdf},
	author = {Stefan Junglen and Harald Luschgy}
}
@article {92,
	title = {Convergence of distributed asynchronous learning vector quantization algorithms},
	year = {2010},
	abstract = {Motivated by the problem of effectively executing clustering algorithms on very large data sets, we address a model for large scale distributed clustering methods. To this end, we briefly recall some standards on the quantization problem and some results on the almost sure convergence of the Competitive Learning Vector Quantization (CLVQ) procedure. A general model for linear distributed asynchronous algorithms well adapted to several parallel computing architectures is also discussed. Our approach brings together this scalable model and the CLVQ algorithm, and we call the resulting technique the Distributed Asynchronous Learning Vector Quantization algorithm (DALVQ). An in-depth analysis of the almost sure convergence of the DALVQ algorithm is performed. A striking result is that we prove that the multiple versions of the quantizers distributed among the processors in the parallel architecture asymptotically reach a consensus almost surely. Furthermore, we also show that these versions converge almost surely towards the same nearly optimal value for the quantization criterion.},
	keywords = {asynchronous, distributed, distributed consensus, k-means, scalability, stochastic optimization, vector quantization},
	attachments = {http://quantif.maths-fi.com/sites/default/files/AsyncStochGradient.pdf},
	author = {Beno{\^\i}t Patra}
}
@article {PagesSellamiSDE,
	title = {Convergence of multi-dimensional quantized SDE{\textquoteright}s},
	year = {2010},
	abstract = {We quantize a multidimensional SDE (in the Stratonovich sense) by solving the related system of ODE{\textquoteright}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 $\frac{1}{q}$-H{\"o}lder distance, $q \superior 2$, in $L^p(\mathbb{P})$.},
	keywords = {functional quantization, H{\"o}lder semi-norm, It{\^o} map, p-variation, rough path theory, stationary quantizers, stochastic differential equations, Stratonovich stochastic integral},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Pages_Sellami_Quantized_SDE_0.pdf},
	author = {Afef Sellami and Gilles Pag{\`e}s}
}
@article {86,
	title = {CVaR hedging using quantization based stochastic approximation algorithm},
	year = {2010},
	abstract = {In this paper, we investigate a method based on risk minimization to hedge observable but non-tradable source of risk on financial or energy markets. The optimal portfolio strategy is obtained by minimizing dynamically the Conditional Value-at-Risk (CVaR) using three main tools: stochastic approximation algorithm, optimal quantization and variance reduction techniques (importance sampling (IS) and linear control variable (LCV)) as the quantities of interest are naturally related to rare events. As a first step, we investigate the problem of CVaR regression, which corresponds to a static portfolio strategy where the number of units of each tradable assets is fixed at time 0 and remains unchanged till time $T$. We devise a stochastic approximation algorithm and study its a.s. convergence and rate of convergence. Then, we extend to the dynamic case under the assumption that the process modelling the non-tradable source of risk and financial assets prices are Markov. Finally, we illustrate our approach by considering several portfolios in the incomplete energy market.},
	keywords = {CVaR, quantization, Robbins-Monro algorithm, stochastic approximation, VaR},
	attachments = {http://quantif.maths-fi.com/sites/default/files/CVaR_hedging.pdf},
	author = {Olivier Bardou and Noufel Frikha and Gilles Pag{\`e}s}
}
@article {51,
	title = {Fractal functional quantization of mean-regular stochastic processes},
	journal = {Mathematical Proceedings of the Cambridge Philosophical Society},
	year = {2010},
	abstract = {We investigate the functional quantization problem for stochastic processes with respect to $L^p(\mathbb{R}^d, \mu)$-norms, where $\mu$ is a fractal measure namely, $\mu$ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of $\mu$ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures $\mu$ we establish a (nonconstructive) link between the quantization errors of $\mu$ and the functional quantization errors of the process in the space $L^p(\mathbb{R}^d, \mu)$.},
	author = {Harald Luschgy and Siegfried Graf and Gilles Pag{\`e}s}
}
@article {CorlayPagesStratification,
	title = {Functional quantization based stratified sampling methods},
	year = {2010},
	abstract = {In this article, we propose several quantization based stratified sampling methods to reduce the variance of a Monte-Carlo simulation. Theoretical aspects of stratification lead to a strong link between the problem of optimal $L^2$-quantization of a random variable and the variance reduction that can be achieved. We first emphasize on the consistency of quantization for designing strata in stratified sampling methods in both finite dimensional and infinite dimensional frameworks. We show that this strata design has a uniform efficiency among the class of Lipschitz continuous functionals. Then a stratified sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multi-factor diffusions. The method is also available for processes made of other Gaussian processes as the Brownian bridge or an Ornstein-Uhlenbeck process. The balance between the algorithmic complexity of the simulation and the variance reduction factor has also been studied.},
	keywords = {Brownian bridge, Brownian motion, functional quantization, Gaussian process, Karhunen-Lo{\`e}ve basis, Monte-Carlo simulation, numerical integration, option pricing, principal component analysis, product quantizer, stratification, Variance reduction, vector quantization, Voronoi diagram},
	attachments = {http://quantif.maths-fi.com/sites/default/files/functional_stratif.pdf},
	author = {Sylvain Corlay and Gilles Pag{\`e}s}
}
@article {61,
	title = {The high resolution vector quantization problem with Orlicz norm distortion},
	year = {2010},
	abstract = {Summary. We derive a high-resolution formula for the quantization problem under Orlicz norm distortion. In this setting, the optimal point density solves a variational problem which comprises a function $g : \mathbb{R_+} \to [0,\infty)$ characterizing the quantization complexity of the underlying Orlicz space. Moreover, asymptotically optimal codebooks induce a tight sequence of empirical measures. The set of possible accumulation points is characterized and in most cases it consists of a single element. In that case, we find convergence as in the classical setting.},
	keywords = {complexity, discrete approximation, High-resolution quantization, self similarity},
	author = {Steffen Dereich and Christian Vormoor}
}
@article {DualQuantizationFoundation,
	title = {Intrinsic stationarity for vector quantization: Foundation of dual quantization},
	year = {2010},
	abstract = {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 for this Delaunay -- or dual -- quantization procedure. We also provide a stochastic optimization method to compute such optimal grids, here for higher dimensional uniform and normal distributions. A crucial feature of this new approach is the fact that it automatically leads to a second order quadrature formula for computing expectations, regardless of the optimality of the underlying grid.},
	keywords = {Delaunay triangulation, numerical integration, quantization, Stationarity, Voronoi tessellation},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Foundation of dual quantization.pdf},
	author = {Gilles Pag{\`e}s and Benedikt Wilbertz}
}
@article {90,
	title = {Numerical method for optimal stopping of piecewise deterministic Markov processes},
	year = {2010},
	abstract = {We propose a numerical method to approximate the value function for the optimal stopping problem of a piecewise deterministic Markov process (PDMP). Our approach is based on quantization of the post jump location - inter-arrival time Markov chain naturally embedded in the PDMP, and path-adapted time discretization grids. It allows us to derive bounds for the convergence rate of the algorithm and to provide a computable $\epsilon$-optimal stopping time. The paper is illustrated by a numerical example.},
	keywords = {dynamic programming, numerical method, Optimal Stopping, piecewise deterministic Markov processes, quantization},
	author = {Beno{\^\i}te de Saporta and Fran{\c c}ois Dufour and Karen Gonzalez}
}
@article {CorlayNystrom,
	title = {The Nystr{\"o}m method for functional quantization with an application to the fractional Brownian motion},
	year = {2010},
	abstract = {In this article, the so-called "Nystr{\"o}m method" is tested to compute optimal quantizers of Gaussian processes. In particular, we derive the optimal quantization of the fractional Brownian motion by approximating the first terms of its Karhunen-Lo{\`e}ve decomposition. 
A numerical test of the "functional stratification" variance reduction algorithm is performed with the fractional Brownian motion.},
	keywords = {Brownian bridge, Brownian motion, fractional Brownian motion, functional quantization, Gaussian process, Gaussian semi-martingale, integral equation, Karhunen-Lo{\`e}ve basis, numerical integration, Nystr{\"o}m method, optimal quantization, Ornstein-Uhlenbeck, product quantization, stratification, Variance reduction, vector quantization},
	attachments = {http://quantif.maths-fi.com/sites/default/files/nystrom_quantization.pdf},
	author = {Sylvain Corlay}
}
@article {82,
	title = {Pricing of barrier options by marginal functional quantization},
	year = {2010},
	abstract = {This paper is devoted to the pricing of Barrier options by optimal quadratic quantization method. From a known useful representation of the premium of barrier options one deduces an algorithm similar to one used to estimate nonlinear filter using quadratic optimal functional quantization. Some numerical tests are fulfilled in the Black-Scholes model and in a local volatility model and a comparison to the so called Brownian Bridge method is also done.},
	keywords = {barrier options, functional quantization, local volatility model, regular Brownian bridge method},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Pricing barrier options by marginal functional quantization.pdf},
	author = {Abass Sagna}
}
@article {94,
	title = {Quantization based filtering method using first order approximation},
	journal = {SIAM journal on numerical analysis},
	volume = {47},
	year = {2010},
	pages = {4711-4734 },
	abstract = {The quantization based filtering method (see <bib>PagesPhamPrintemsSurvey</bib> is a grid based approximation method to solve nonlinear filtering problems with discrete time observations. It relies on off-line preprocessing of some signal grids in order to construct fast recursive schemes for filter approximation. We give here an improvement of this method by taking advantage of the stationary quantizer property. The key ingredient is the use of vanishing correction terms to describe schemes based on piecewise linear approximations. Convergence results are given and numerical results are presented for the particular cases of linear Gaussian model and stochastic volatility models.},
	keywords = {approximation order, convergence, discrete time, Filtering, finance, grid pattern, linear approximation, linear model, Monte Carlo method, non linear approximation, numerical analysis, numerical approximation, numerical integration, numerical method, stochastic method, stochastic model},
	author = {Afef Sellami}
}
@article {83,
	title = {Sharp rate for the dual quantization problem},
	year = {2010},
	abstract = {In this paper we establish the sharp rate of the optimal dual quantization problem. The notion of dual quantization was recently introduced in the paper [bib]DualQuantizationFoundation[/bib], where it was shown that, at least in an Euclidean setting, dual quantizers are based on a Delaunay triangulation, the dual counterpart of the Voronoi tessellation on which "regular" quantization relies. Moreover, this new approach 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. The proof of this theorem relies among others on an extension of the so-called Pierce lemma by means of a random quantization argument.},
	keywords = {Delaunay triangulation, dual quantization, Pierce{\textquoteright}s lemma, quantization, random quantization, rate, Zador{\textquoteright}s theorem},
	author = {Gilles Pag{\`e}s and Benedikt Wilbertz}
}
@article {47,
	title = {A variance reduction technique using a quantized Brownian motion as a control variate},
	year = {2010},
	abstract = {This article presents a new variance reduction technique for diffusion processes where a control variate is constructed using a quantization of the coefficients of the Karhunen-Lo{\`e}ve decomposition of the underlying Brownian motion. This method may be indeed used for other Gaussian processes.},
	keywords = {Variance reduction},
	author = {Antoine Lejay and Victor Reutenauer}
}
@article {67,
	title = {An application to credit risk of a hybrid Monte Carlo-Optimal quantization method},
	year = {2009},
	abstract = {In this paper we use a hybrid Monte Carlo-Optimal quantization method to approximate the conditional survival probabilities of a firm, given a structural model for its credit defaul, under partial information. We consider the case when the firm{\textquoteright}s value is a non-observable stochastic process $(V_t)_{t \geq 0}$ and inverstors in the market have access to a process $(S_t)_{t \geq 0}$, whose value at each time $t$ is related to $(V_s, s \leq t)$. We are interested in the computation of the conditional survival probabilities of the firm given the "investor information". As a application, we analyse the shape of the credit spread curve for zero coupon bonds in two examples. },
	keywords = {computational finance, credit risk, Filtering, Monte-Carlo method, optimal quantization, partial information, probability, quantitative finance, risk management, structural approach, survival probabilities},
	author = {Giorgia Callegaro and Abass Sagna}
}
@article {43,
	title = {Dual Quantization for random walks with application to credit derivatives},
	year = {2009},
	abstract = {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{\textquoteright}s method.},
	author = {Gilles Pag{\`e}s and Benedikt Wilbertz}
}
@article {62,
	title = {High resolution quantization and entropy coding of jump processes},
	journal = {Journal of Complexity},
	volume = {25},
	year = {2009},
	pages = {163-187},
	abstract = {We study the quantization problem for certain types of jump processes. The probabilities for the number of jumps are assumed to be bounded by Poisson weights. Otherwise, jump positions and increments can be rather generally distributed and correlated. We show in particular that in many cases entropy coding error and quantization error have distinct rates. Finally, we investigate the quantization problem for the special case of $\mathbb{R}^d$-valued compound Poisson processes.},
	keywords = {complexity, compound Poisson process, entropy coding, High resolution quantization, jump process, L{\'e}vy process, metric entropy},
	author = {Frank Aurzada and Steffen Dereich and Michael Scheutzow and Christian Vormoor}
}
@article {6,
	title = {Optimal quantization for the pricing of swing options},
	journal = {Applied Mathematical Finance},
	volume = {16},
	year = {2009},
	note = {Comment: 27p},
	pages = {183-217},
	abstract = {In this paper, we investigate a numerical algorithm for the pricing of swing options, relying on the so-called optimal quantization method. The numerical procedure is described in details and numerous simulations are provided to assert its efficiency. In particular, we carry out a comparison with the Longstaff-Schwartz algorithm.},
	author = {Olivier Bardou and Sandrine Bouthemy and Gilles Pag{\`e}s}
}
@article {52,
	title = {Quantization for probability measures in the Prokhorov metric},
	journal = {Theory Probab. Appl.},
	volume = {53},
	year = {2009},
	pages = {216-241},
	abstract = {For a probability distribution $P$ on $\mathbb{R}^d$ and $n\in N$ consider $e_n=inf \pi(P,Q)$, where $\pi$ denotes the Prokhorov metric and the infimum is taken over all discrete probabilities $Q$ with $|\textrm{supp}(Q)|\leq n$. We study solutions $Q$ of this minimization problem, stability properties, and consistency of empirical estimators. For some classes of distributions we determine the exact rate of convergence to zero of the $n$th quantization error en as $n\to \infty$.	},
	keywords = {asymptotic quantization error, empirical measures, entropy, Ky Fan metric, multidimensional quantization, optimal quantizers, Prokhorov metric, quantization dimension},
	author = {Harald Luschgy and Siegfried Graf}
}
@article {68,
	title = {Universal $L^s$-rate-optimality of $L^r$-optimal quantizers by dilatation and contraction},
	journal = {ESAIM: Probability and Statistics},
	volume = {13},
	year = {2009},
	pages = {218-246},
	abstract = {We investigate in this paper the properties of some dilatations or contractions of a sequence $(\alpha _{n})_{n \ge 1}$ of $L^r$-optimal quantizers of an $\mathbb{R}^d$-valued random vector $X \in L^r(\mathbb{P})$ defined in the probability space $(\Omega ,\mathcal{A},\mathbb{P})$ with distribution $\mathbb{P}_{X} = P$. To be precise, we investigate the $L^s$-quantization rate of sequences $\alpha _n^{\theta ,\mu } = \mu + \theta (\alpha _n-\mu )=\lbrace \mu + \theta (a-\mu ), \ a \in \alpha _n \rbrace $ when $\theta \in \mathbb{R}_{+}^{\star }, \mu \in \mathbb{R}, s \in (0,r)$ or $s \in (r,+\infty )$ and $X \in L^s(\mathbb{P})$. We show that for a wide family of distributions, one may always find parameters $(\theta ,\mu )$ such that $(\alpha _n^{\theta ,\mu })_{n \ge 1}$ is $L^s$-rate-optimal. For the gaussian and the exponential distributions we show the existence of a couple $(\theta ^{\star }, \mu ^{\star })$ such that $(\alpha ^{\theta ^{\star },\mu ^{\star }})_{n \ge 1}$ also satisfies the so-called $L^s$-empirical measure theorem. Our conjecture, confirmed by numerical experiments, is that such sequences are asymptotically $L^s$-optimal. In both cases the sequence $(\alpha ^{\theta ^{\star },\mu ^{\star }})_{n \ge 1}$ is incredibly close to $L^s$-optimality. However we show (see Rem. 5.4) that this last sequence is not $L^s$-optimal (e.g. when $s = 2$, $r = 1$) for the exponential distribution. },
	keywords = {dilatation, empirical measure theorem, Lloyd algorithm, rate-optimal quantizers},
	author = {Abass Sagna}
}
@article {DereichDiffusionCoding,
	title = {The coding complexity of diffusion processes under Lp[0, 1]-norm distortion},
	journal = {Stochastic Processes and their Applications},
	volume = {118},
	year = {2008},
	month = {June},
	pages = {938-951},
	abstract = {We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under $L^p[0, 1]$-norm distortion. We find explicit high resolution formulas in terms of the average diffusion coefficient seen by the process. The proof is based on a decoupling method introduced in a former article by the author. Given that link it remains to analyze the coding problem for a concatenation of Wiener processes and to solve the corresponding rate allocation problem.},
	keywords = {complexity, High-resolution quantization, stochastic differential equations, stochastic processes},
	author = {Steffen Dereich}
}
@booklet {13,
	title = {Asymptotics of the maximal radius of an ${L}^r$-optimal sequence of quantizers},
	year = {2008},
	abstract = {Let $P$ be a probability distribution on $\mathbb{R}^d$ (equipped with an Euclidean norm). Let $r, s \superior 0$ and assume $(\alpha_n)_{n \geq 1}$ is an (asymptotically) $L^r(P)$-optimal sequence of $n$-quantizers. In this paper we investigate the asymptotic behavior of the maximal radius sequence induced by the sequence $(\alpha_n)_{n \geq 1}$ and defined to be for every $n \geq 1$, $\ \rho(\alpha_n) = \max \{| a |, a \in \alpha_n \}$. We show that if ${\rm card(supp}(P))$ is infinite, the maximal radius sequence goes to $\sup \{| x |, x \in {\rm supp}(P) \}$ as $n$ goes to infinity. We then give the rate of convergence for two classes of distributions with unbounded support : distributions with exponential tails and distributions with polynomial tails. },
	author = {Gilles Pag{\`e}s and Abass Sagna}
}
@article {93,
	title = {Comparative survey on nonlinear filtering methods: the quantization and the particle filtering approaches},
	journal = {Journal of Statistical Computation and Simulation},
	volume = {78},
	year = {2008},
	pages = {93-113},
	abstract = {We provide a comparative study between two different approaches to construct nonlinear filter estimators: on the one hand grid methods using zero-order and first-order quantization schemes, and on the other hand particle filtering algorithms using sequential importance sampling or resampling. For each method, numerical implementation is explicited in addition to convergence arguments and algorithmic complexity. Numerical examples are then given over three state space models: the Kalman filter case, the canonical stochastic volatility model and the infinite dimension explicit filter introduced in [Genon-Catalot, V., 2003, A non linear explicit filter.},
	keywords = {importance sampling, infinite dimension filter, Kalman filter, nonlinear filter, particle filtering, quantization schemes, stochastic volatility},
	author = {Afef Sellami}
}
@mastersthesis {44,
	title = {Construction of optimal quantizers for Gaussian measures on Banach spaces},
	year = {2008},
	school = {Universit{\"a}t Trier},
	abstract = {In this thesis, we investigate the quantization problem of Gaussian measures on Banach spaces by means of constructive methods.
That is, for a random variable $X$ and a natural number $N$, we are searching for those $N$ elements in the underlying Banach space which give the best approximation to $X$ in the average sense.
We particularly focus on centered Gaussians on the space of continuous functions on $[0,1]$ equipped with the supremum-norm, since in that case all known methods failed to achieve the optimal quantization rate for important Gauss-processes.
In fact, by means of Spline-approximations and a scheme based on the Best-Approximations in the sense of the Kolmogorov $n$-width we were able to attain the optimal rate of convergence to zero for these quantization problems.
Moreover, we established a new upper bound for the quantization error, which is based on a very simple criterion, the modulus of smoothness of the covariance function.
Finally, we explicitly constructed those quantizers numerically.},
	keywords = {approximation, Banach space, Brownian motion, Gaussian measures, quantization},
	author = {Benedikt Wilbertz}
}
@article {53,
	title = {Functional quantization rate and mean pathwise regularity of processes with an application to L{\'e}vy processes},
	journal = {Annals of Applied Probability},
	volume = {18},
	year = {2008},
	pages = {427-469},
	abstract = {We investigate the connections between the mean pathwise regularity of stochastic processes and their $L^r(P)$-functional quantization rates as random variables taking values in some $L^p([0,T],dt)$-spaces ($0 < p \leq r$). Our main tool is the Haar basis. We then emphasize that the derived functional quantization rate may be optimal (e.g., for Brownian motion or symmetric stable processes) so that the rate is optimal as a universal upper bound. As a first application, we establish the $O((\log N)^{-1/2})$ upper bound for general It{\^o} processes which include multidimensional diffusions. Then, we focus on the specific family of L{\'e}vy processes for which we derive a general quantization rate based on the regular variation properties of its L{\'e}vy measure at $0$. The case of compound Poisson processes, which appear as degenerate in the former approach, is studied specifically: we observe some rates which are between the finite-dimensional and infinite-dimensional "usual{\textquoteright}{\textquoteright} rates},
	author = {Harald Luschgy and Gilles Pag{\`e}s}
}
@article {46,
	title = {How to speed up the quantization tree algorithm with an application to swing options},
	year = {2008},
	abstract = {In this paper, we suggest several improvements to the numerical implementation of the quantization method for stochastic control problems in order to get fast and accurate premium estimations. This technique is applied to derivative pricing in energy markets. Several ways of modeling energy derivatives are described and finally numerical examples including parallel execution on multi-processor devices are presented to illustrate the accuracy of these methods and their execution times.},
	keywords = {Backward Dynamic programming, Gaussian process, L{\'e}vy process, quantization, Stochastic control, Swing options},
	author = {Anne Laure Bronstein and Gilles Pag{\`e}s and Benedikt Wilbertz}
}
@mastersthesis {69,
	title = {M{\'e}thodes de quantification optimale avec applications {\`a} la finance},
	year = {2008},
	abstract = {This thesis is dovoted to optimal quantization with some applications to mathematical finance. Chap.1 reminds the bases of optimal quantization and numerical search of optimal quantizers. In chap.2 we study the asymptotics, in $L^s$, of the quantization error associated to a linear transform of an $L^r$ optimal sequence of quantizers. We show that such a transformation allows to make the transformed sequence $L^s$ rate optimal for every $s>0$, for a large family of probabilities. Chap.3 deals with the asymptotics of the maximal radius sequence associated to an $L^r$ optimal sequence of quantizers. We show that as soon as $\textrm{supp}(p)$ is unbounded, the maximal radius converge to infinity. We then give the rate of convergence for a large family of probabilities. Chap.4 is devoted to the pricing of lookback and barrier like options. We write these prices in a form which allows us to estimate them by Monte-Carlo, by an hybrid Monte-Carlo-quantization and by a pure quantization method. },
	keywords = {barrier options, Lloyd algorithm, lookback options, maximal radius, quantization, rate-optimal},
	author = {Abass Sagna}
}
@article {Handbook_Finance,
	title = {Optimal quantization for Finance: from random vectors to stochastic processes},
	journal = {Handbook of Numerical Analysis},
	volume = {15},
	year = {2008},
	abstract = {We review optimal quantization methods for numerically solving nonlinear problems in higher dimension associated with Markov processes. Quantization of a Markov process consists in a spatial discretization on finite grids optimally fitted to the dynamicsof the process. Two quantization methods are proposed: the first one, called marginal quantization, relies on an optimal approximation of the marginal distributions of the process, while the second one, called Markovian quantization, looks for an optimal approximation of transition probabilities of the Markov process at some points. Optimal grids and their associated weights can be computed by a stochastic gradient descent method based on Monte Carlo simulations. We illustrate this optimal quantization approach with four numerical applications arising in finance: European option pricing, optimal stopping problems and American option pricing, stochastic control problems and mean-variance hedging of options and filtering in stochastic volatility models. },
	keywords = {Euler scheme, Markov chain, non linear filtering, numerical integration, Optimal Stopping, option pricing, quantization, Stochastic control, stochastic gradient descent},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Handbook_2008_Quantization_in_finance.pdf},
	author = {Gilles Pag{\`e}s and Jacques Printems}
}
@article {65,
	title = {Linear transformations and the k-means clustering algorithm: applications to clustering curves},
	journal = {Am Stat.},
	volume = {61(1)},
	year = {2007},
	pages = {34{\textendash}40},
	abstract = {Functional data can be clustered by plugging estimated regression coefficients from individual curves into the k-means algorithm. Clustering results can differ depending on how the curves are fit to the data. Estimating curves using different sets of basis functions corresponds to different linear transformations of the data. k-means clustering is not invariant to linear transformations of the data. The optimal linear transformation for clustering will stretch the distribution so that the primary direction of variability aligns with actual differences in the clusters. It is shown that clustering the raw data will often give results similar to clustering regression coefficients obtained using an orthogonal design matrix. Clustering functional data using an $L^2$ metric on function space can be achieved by clustering a suitable linear transformation of the regression coefficients. An example where depressed individuals are treated with an antidepressant is used for illustration.},
	author = {Thaddeus Tarpey}
}
@article {23,
	title = {Optimal quantizers for Radon random vectors in a Banach space},
	journal = {J. Approx. Theory},
	volume = {144},
	number = {1},
	year = {2007},
	pages = {27{\textendash}53},
	publisher = {Academic Press, Inc.},
	address = {Orlando, FL, USA},
	abstract = {For every integer $n$ and evrery positive real number $r \superior 0$ and a Radon random vector $X$ with values in a Banach space $E$, let $e_{n,r}(X,E) = \inf{\left(E\left[\min\limits_{a \in \alpha} \| X-a \|^r \right]^{1/r}\right)}$, where the infimum is taken over all subsets $\alpha$ of $E$ with $\textrm{card}(\alpha) \leq n$ ($n$-quantizers). We investigate the existence of optimal $n$-quantizers for this $L^r$-quantization problem, derive their stationarity properties and establish for $L^p$-spaces $E$ the pathwise regularity of stationary quantizers.},
	issn = {0021-9045},
	author = {Siegfried Graf and Harald Luschgy and Gilles Pag{\`e}s}
}
@booklet {14,
	title = {Quadratic optimal functional quantization of stochastic processes and numerical applications},
	year = {2007},
	note = {Comment: 41 pages},
	abstract = {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 Hilbert-valued 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.},
	author = {Gilles Pag{\`e}s}
}
@article {50,
	title = {A forward{\textendash}backward stochastic algorithm for quasi-linear PDEs},
	journal = {Annals of Applied Probability},
	volume = {16(1)},
	year = {2006},
	pages = {140-184},
	abstract = {We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward{\textendash}backward SDEs, which provides an efficient probabilistic representation of this type of equation. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE. In particular, our work provides an alternative to the method described in [Douglas, Ma and Protter (1996) Ann. Appl. Probab. 6  940{\textendash}968] and weakens the regularity assumptions required in this reference. },
	author = {Fran{\c c}ois Delarue and St{\'e}phane Menozzi}
}
@article {25,
	title = {Functional quantization of a class of Brownian diffusions: a constructive approach},
	journal = {Stochastic Processes and their Applications},
	volume = {116},
	number = {2},
	year = {2006},
	pages = {310 - 336},
	abstract = {The functional quantization problem for one-dimensional Brownian diffusions on $[0,T]$ is investigated. One shows under rather general assumptions that the rate of convergence of the $L^p$-quantization error is $O(\log(n)^{-1/2})$ like for the Brownian motion. Several methods to construct some rate-optimal quantizers are proposed. These results are extended to $d$-dimensional diffusions when the diffusion coefficient is the inverse of a gradient function. Finally, a special attention is given to diffusions with a Gaussian martingale term.},
	keywords = {Brownian diffusions, functional quantization, Lamperti transform; Girsanov theorem, optimal quantizers},
	issn = {0304-4149},
	author = {Harald Luschgy and Gilles Pag{\`e}s}
}
@article {30,
	title = {Optimal quantization : evolutionary algorithm vs stochastic gradient},
	journal = {Proceedings of the 9th Joint Conference on Information Sciences},
	year = {2006},
	publisher = {Atlantic Press},
	abstract = {We propose a new method based on evolutionary optimization for obtaining an optimal $L^p$-quantizer of a multidimensional random variable. First, we remind briefly the main results about quantization. Then, we present the classical gradient-based approach (this approach is well detailed in [2] and [7] for $p=2$) used up to now to find a {\textquotedblleft}local{\textquotedblright} optimal $L^p$-quantizer. Then, we give an algorithm that permits to deal with the problem in the evolutionary optimization framework and illustrate a numerical comparison between the proposed method and the stochastic gradient method. Finally, a numerical application to option pricing in finance is provided.},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Evolutionary-Algorithms.pdf},
	author = {Moez Mrad and Sana Ben Hamida}
}
@booklet {37,
	title = {Pricing path-dependent options using optimized functional quantization},
	journal = {Numerical Methods in Finance},
	year = {2006},
	keywords = {Asian option, Brownian motion, functional quantization, Gaussian process, Heston model, numerical integration, optimal quantization},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Pricing Path Dependent Option Using Functional Quantization.pdf},
	author = {Gilles Pag{\`e}s and Jacques Printems}
}
@article {18,
	title = {Quantization of probability distributions under norm-based distortion measures II: Self-similar distributions},
	journal = {Journal of Mathematical Analysis and Applications},
	volume = {318},
	number = {2},
	year = {2006},
	pages = {507 - 516},
	abstract = {For a probability measure $P$ on $\mathbb{R}^d$ and $n \in  \mathbb{N}$ consider $e_n = \inf \displaystyle \int \min_{a \in \alpha} V(\| x-a \| )dP(x)$ where the infimum is taken over all subsets $\alpha$ of $\mathbb{R}^d$ with $\mbox{card} (\alpha) \leq n$ and $V$ is a nondecreasing function. Under certain conditions on $V$, we derive the precise $n$-asymptotics of $e_n$ for nonsingular and for (singular) self-similar distributions $P$ and we find the asymptotic performance of optimal quantizers using weighted empirical measures.},
	keywords = {Point density measure},
	issn = {0022-247X},
	author = {Sylvain Delattre and Siegfried Graf and Harald Luschgy and Gilles Pag{\`e}s}
}
@article {36,
	title = {Approximation by quantization of the filter process and applications to optimal stopping problems under partial observation},
	journal = {Monte Carlo methods and Applications},
	volume = {11(1)},
	year = {2005},
	pages = {57-81},
	abstract = {We present an approximation method for discrete time nonlinear filtering in view of solving dynamic optimization problems under partial information. The method is based on quantization of the Markov pair process filter-observation $(\Pi, Y )$ and is such that, at each time step k and for a given size $N_k$ of the quantization grid in period $k$, this grid is chosen to minimize a suitable quantization error. The algorithm is based on a stochastic gradient descent combined with Monte-Carlo simulations of $(\Pi, Y )$. Convergence results are given and applications to optimal stopping under partial observation are discussed. Numerical results are presented for a particular stopping problem : American option pricing with unobservable volatility.},
	keywords = {Markov chain, Monte-Carlo simulations, Nonlinear filtering, Optimal Stopping, partial observation, quantization, stochastic gradient descent},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Quantization filter process optimal stopping.pdf},
	author = {Huy{\^e}n Pham and Wolfgang Runggaldier and Afef Sellami}
}
@article {34,
	title = {Discretization and simulation for a class of SPDEs with applications to Zakai and McKean-Vlasov equations},
	year = {2005},
	note = {60{H}35, 60{H}15, 60{G}35, 60{H}07, 65{C}20.},
	abstract = {This paper is concerned with numerical approximations for a class of nonlinear stochastic partial differential equations: Zakai equation of nonlinear filtering problem and McKean-Vlasov type equations. The approximation scheme is based on the representation of the solutions as weighted conditional distributions. We first accurately analyse the error caused by an Euler type scheme of time discretization. Sharp error bounds are calculated: we show that the rate of convergence is in general of order $\sqrt{\Delta}$ ( $\Delta$ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes $\Delta$. This result is obtained by carefully employing techniques of Malliavin calculus. In a second step, we propose a simulation of the time discretization Euler scheme by a quantization approach. This formally consists in an approximation of the weighted conditional distribution by a conditional discrete distribution on finite supports. We provide error bounds and rate of convergence in terms of the number $N$ of the grids of this support. These errors are minimal at some optimal grids which are computed by a recursive method based on Monte Carlo simulations. Finally, we illustrate our results with some numerical experiments arising from correlated Kalman-Bucy filter and Burgers equation.},
	keywords = {Euler scheme, Malliavin calculus, Malliavin calculus., McKean-Vlasov equations, Nonlinear filtering, quantization, Stochastic partial differential equations, {E}uler scheme, {M}alliavin calculus., {M}c{K}ean-{V}lasov equations, {N}onlinear filtering, {S}tochastic partial differential equations},
	author = {Emmanuel Gobet and Gilles Pag{\`e}s and Huy{\^e}n Pham and Jacques Printems}
}
@article {54,
	title = {Entropy-constrained functional quantization of Gaussian measures},
	journal = {Proceedings AMS},
	volume = {133},
	year = {2005},
	pages = {3403-3409 },
	abstract = {The sharp asymptotics for the entropy-constrained $L^2$-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 of the covariance operator.
},
	author = {Harald Luschgy and Siegfried Graf}
}
@article {FunctionalQuantizationNum,
	title = {Functional quantization for numerics with an application to option pricing},
	journal = {Monte Carlo Methods and Appl.},
	volume = {11},
	number = {11},
	year = {2005},
	pages = {407-446},
	abstract = {    We investigate in this paper the numerical performances of quadratic functional quantization with some applications to Finance. We emphasize the r{\^o}le played by the so-called product quantizers and the Karhunen-Lo{\`e}ve expansion of Gaussian processes, in particular the Brownian motion. We show how to build some efficient functional quantizers for Brownian diffusions. We propose a quadrature formula based on a Romberg log-extrapolation of {\textquotedblleft}crude{\textquotedblright} functional quantization which speeds up significantly the method. Numerical experiments are carried out on two European option pricing problems: vanilla and Asian Call options in a Heston stochastic volatility model. It suggests that functional quantization is a very efficient integration method for various path-dependent functionals of a diffusion processes: it produces deterministic results which outperforms Monte Carlo simulation for usual accuracy levels. },
	keywords = {Asian option, Brownian motion, functional quantization, Heston model, Karhunen-Lo{\`e}ve expansion, product quantizers, Romberg extrapolation, SDE, stochastic volatility},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Functional Quantization Num.pdf},
	author = {Gilles Pag{\`e}s and Jacques Printems}
}
@booklet {24,
	title = {High-resolution product quantization for Gaussian processes under sup-norm distortion},
	year = {2005},
	abstract = {We derive high-resolution upper bounds for optimal product quantization of pathwise contionuous Gaussian processes respective to the supremum norm on $[0,T]^d$. Moreover, we describe a product quantization design which attains this bound. This is achieved under very general assumptions on random series expansions of the process. It turns out that product quantization is asymptotically only slightly worse than optimal functional quantization. The results are applied e.g. to fractional Brownian sheets and the Ornstein-Uhlenbeck process.},
	keywords = {Gaussian process, High-resolution quantization, product quantization, series expansion},
	author = {Harald Luschgy and Gilles Pag{\`e}s}
}
@mastersthesis {SellamiPhD,
	title = {M{\'e}thodes de quantification optimale pour le filtrage et applications {\`a} la finance},
	year = {2005},
	school = {Universit{\'e} Paris Dauphine},
	abstract = {We develop a grid based numerical approach to solve a filtering problem, using results on optimal quantization of random variables. We construct two filtering algorithms using zero order and first order approximation techniques. We suggest implementable versions of these algorithms and study the approximation error behavior by considering the stationnary property of optimal quantizers. The grid approach is then compared to the particle one based on Monte Carlo methods. The study is done over a set of different state models. In a second part, we have been interested in the advantadge given by quantization methods to preprocess offline the information. This permitted to develop a filtering algorithm based on observation (and signal) quantization. Here also the error convergence rate to zero as the quantizer size goes to infinity is studied. Finally, the quantization of the filter as a random variable is studied in order to solve a problem of pricing an American option in an unobserved stochastic volatility market. All results are illustrated by numerical experiments},
	keywords = {American option, Filtering, Monte Carlo, offline preprocessing, optimal quantization, particle method, stationary quantizer, stochastic volatility},
	attachments = {http://quantif.maths-fi.com/sites/default/files/PhD_AfefSELLAMI.pdf , http://quantif.maths-fi.com/sites/default/files/Introduction_Sellami.pdf},
	author = {Afef Sellami}
}
@article {PagesPhamNonLinearFiltering,
	title = {Optimal quantization methods for nonlinear filtering with discrete-time observations},
	journal = {Bernoulli},
	volume = {11(5)},
	year = {2005},
	abstract = {We develop an optimal quantization approach for numerically solving nonlinear filtering problems associated with discrete-time or continuous-time state process and discrete-time observations. Two quantization methods are proposed: a marginal quantization and a Markovian quantization of the signal process. The approximate filters are explicitly solved by a finite-dimensional forward procedure. A posteriori error bounds are stated and we show that the approximate error terms are minimal at some specific grids that may be computed off-line by a stochastic gradient method based on Monte Carlo simulations. Some numerical experiments are carried out: the convergence of the approximate filter as the accuracy of the quantization increases and its stability when the latent process is mixing are emphasized.},
	keywords = {Euler scheme, Markov chain, nonlinear filtering, numerical approximation, stationary signal, stochastic gradient descent, vector quantization},
	author = {Gilles Pag{\`e}s and Huy{\^e}n Pham}
}
@article {55,
	title = {The point density measure in the quantization of self-similar probabilities},
	journal = {Math. Proc. Cambridge Phil. Soc. },
	volume = {138},
	year = {2005},
	pages = {513-531},
	author = {Harald Luschgy and Siegfried Graf}
}
@article {BallyPagesPrintemsAmerican,
	title = {A quantization tree method for pricing and hedging multidimensional American options},
	journal = {Mathematical Finance},
	volume = {15},
	number = {1},
	year = {2005},
	pages = {119-168},
	abstract = {We present here the quantization method which is well-adapted for the pricing and hedging of American options on a basket of assets. Its purpose is to compute a large number of conditional expectations by projection of the diffusion on optimal grid designed to minimize the (square mean) projection error ([24]). An algorithm to compute such grids is described. We provide results concerning the orders of the approximation with respect to the regularity of the payoff function and the global size of the grids. Numerical tests are performed in dimensions 2, 4, 6, 10 with American style exchange options. They show that theoretical orders are probably pessimistic.},
	keywords = {American option pricing, local volatility model, optimal quantization, Optimal Stopping, Snell envelope},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Quantization-Tree.pdf},
	author = {Vlad Bally and Gilles Pag{\`e}s and Jacques Printems}
}
@article {20,
	title = {Local distortion and {\textmu}-mass of the cells of one dimensional asymptotically optimal quantizers},
	journal = {Communications in Statistics. Theory and Methods},
	volume = {33(5)},
	year = {2004},
	month = {2004},
	chapter = {1087-1117},
	abstract = {We consider one dimensional probability distributions $\mu$ having a continuous and positive probability density function. We find the asymptotic of the size and the mass of the Voronoi cells and we prove that the local distortion associated with stationary or optimal quantizers is asymptotically uniform. Numerical simulations and computations illustrate the theoretical results and lead to the design of some good-fit test for the stationary equilibria.},
	author = {Sylvain Delattre and Fort, Jean-Claude and Gilles Pag{\`e}s}
}
@article {OptimalMarkovianQuantizationControl,
	title = {An optimal Markovian quantization algorithm for multidimensional stochastic control problems},
	journal = {Stochastics and Dynamics},
	volume = {4(4)},
	year = {2004},
	pages = {501-545},
	abstract = {We propose a probabilistic numerical method based on optimal quantization to solve some multi-dimensional stochastic control problems that arise, for example, in mathematical finance for portfolio optimization. We then consider some controlled diffusions with most components control free. The Euler scheme of the uncontrolled diffusion part is approximated by a discrete time process obtained by a nearest neighbor projection on some grids optimally fitted to its dynamics. The resulting process is also designed to preserve the Markov property with respect to the filtration of the Euler scheme. This Markovian quantization approach leads to an approximate control problem that can be solved numerically by the dynamic programming formula. This approach seems promising in higher dimension. A priori Lp-error bounds are stated and we show that the spatial discretization error term is minimal at some specific grids. A simple recursive algorithm is devised to compute these optimal grids by induction based on a Monte Carlo simulation. Some numerical illustrations are processed for solving a mean-variance hedging problem.},
	keywords = {Euler scheme, Markov chain, mean-variance hedging, Stochastic control, stochastic gradient descent, vector quantization},
	author = {Gilles Pag{\`e}s and Huy{\^e}n Pham and Jacques Printems}
}
@article {PagesPhamPrintemsSurvey,
	title = {Optimal quantization methods and applications to numerical problems in finance},
	year = {2004},
	abstract = {We review optimal quantization methods for numerically solving nonlinear problems in higher dimension associated with Markov processes. Quantization of a Markov process consists in a spatial discretization on finite grids optimally fitted to the dynamics of the process. Two quantization methods are proposed: the first one, called marginal quantization, relies on an optimal approximation of the marginal distributions of the process, while the second one, called Markovian quantization, looks for an optimal approximation of transition probabilities of the Markov process at some points. Optimal grids and their associated weights can be computed by a stochastic gradient descent method based on Monte Carlo simulations. We illustrate this optimal quantization approach with four numerical applications arising in finance: European option pricing, optimal stopping problems and American option pricing, stochastic control problems and mean-variance hedging of options and filtering in stochastic volatility models.},
	keywords = {Euler scheme, Markov chain, non linear filtering, numerical integration, Optimal Stopping, option pricing, quantization, Stochastic control, stochastic gradient descent},
	author = {Gilles Pag{\`e}s and Huy{\^e}n Pham and Jacques Printems}
}
@article {56,
	title = {Quantization for probability measures with respect to the geometric mean error},
	journal = {Math. Proc. Cambridge Phil. Soc.},
	volume = {136},
	year = {2004},
	pages = {687-717},
	author = {Harald Luschgy and Siegfried Graf}
}
@article {delattre,
	title = {Quantization of probability distributions under norm-based distortion measures},
	year = {2004},
	note = {60{E}99, 94{A}29, 28{A}80},
	abstract = {For a probability measure $P$ on $\mathbb{R}^d$ and $n \in \mathbb{N}$ consider $e_n = \inf  \int \min_{a \in \alpha} V(\| x-a \| )dP(x)$ where the infimum is taken over all subsets $\alpha$ of $\{R}^d$ with $\mbox{card} (\alpha) \leq n$ and ${V}$ is a nondecreasing function. Under certain conditions on $V$, we derive the precise $n$-asymptotics of $e_n$ for nonsingular and for (singular) self-similar distributions $P$ and we find the asymptotic performance of optimal quantizers using weighted empirical measures.},
	keywords = {empirical measure, High-rate vector quantization, local distortion, norm-difference distortion, Point density measure, weak convergence},
	author = {Sylvain Delattre and Siegfried Graf and Harald Luschgy and Gilles Pag{\`e}s}
}
@article {9,
	title = {Sharp asymptotics of the functional quantization problem for Gaussian processes},
	volume = {32},
	number = {2},
	year = {2004},
	note = {Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Probability (http://www.imstat.org/aop/) at http://dx.doi.org/10.1214/009117904000000324},
	abstract = {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. },
	author = {Harald Luschgy and Gilles Pag{\`e}s}
}
@article {19,
	title = {Sharp asymptotics of the Kolmogorov entropy for Gaussian measures},
	journal = {Journal of Functional Analysis},
	volume = {212},
	number = {1},
	year = {2004},
	pages = {89 - 120},
	abstract = {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.},
	keywords = {Shannon entropy},
	issn = {0022-1236},
	author = {Harald Luschgy and Gilles Pag{\`e}s}
}
@article {31,
	title = {Error analysis of the quantization algorithm for obstacle problems},
	journal = {Stochastic Processes \& Their Applications},
	volume = {106(1)},
	year = {2003},
	chapter = {1-40},
	abstract = {In the accompanying paper [2] an algorithm based on a "quantized tree" is designed to compute the solution of multi-dimensional obstacle problems for homogeneous $\mathbb{R}^d$-valued Markov chains. It is based on the quantization of probability distributions which yields a dynamic programming formula on a discrete tree. A typical example of such problems is the pricing of multi-asset American style vanilla options. In the first part of the present paper, the analysis of the $L^p$-error is completed. In the second part, we estimate the error induced by the Monte Carlo estimation of the transition weights involved in the (optimal) quantized tree.},
	keywords = {American option pricing, Numerical Probability, Optimal Stopping, Quantization of random variables, Reflected Backward Stochastic Differential Equation, Snell envelope},
	author = {Vlad Bally and Gilles Pag{\`e}s}
}
@article {33,
	title = {First order schemes in the numerical quantization method},
	journal = {Mathematical Finance},
	volume = {13},
	year = {2003},
	pages = {1-16},
	abstract = {The numerical quantization method is a grid method that relies on the approximation of the solution to a nonlinear problem by piecewise constant functions. Its purpose is to compute a large number of conditional expectations along the path of the associated diffusion process. We give here an improvement of this method by describing a first-order scheme based on piecewise linear approximations. Main ingredients are correction terms in the transition probability weights. We emphasize the fact that in the case of optimal quantization, many of these correcting terms vanish. We think that this is a strong argument to use it. The problem of pricing and hedging American options is investigated and a priori estimates of the errors are proposed. },
	attachments = {http://quantif.maths-fi.com/sites/default/files/First Order Schmes.pdf},
	author = {Vlad Bally and Gilles Pag{\`e}s and Jacques Printems}
}
@article {22,
	title = {Functional quantization and small balls probabilities for Gaussian processes},
	journal = {Journal of Theoretical Probability},
	volume = {16(4)},
	year = {2003},
	chapter = {1047-1062},
	abstract = {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.},
	author = {Siegfried Graf and Harald Luschgy and Gilles Pag{\`e}s}
}
@article {64,
	title = {On the link between small ball probabilities and the quantization problem for Gaussian measures on Banach spaces},
	journal = {Journal of Theoretical Probability},
	volume = {16(1)},
	year = {2003},
	abstract = {Let m be a centered Gaussian measure on a separable Banach space $E$ and $N$ a positive integer. We study the asymptotics as $N \to \infty$ of the quantization error, i.e., the infimum over all subsets $\varepsilon$ of $E$ of cardinality $N$ of the average distance w.r.t. $m$ to the closest point in the set $\varepsilon$. We compare the quantization error with the average distance which is obtained when the set $E$ is chosen by taking $N$ i.i.d. copies of random elements with law $m$. Our approach is based on the study of the asymptotics of the measure of a small ball around $0$. Under slight conditions on the regular variation of the small ball function, we get upper and lower bounds of the deterministic and random quantization error and are able to show that both are of the same order. Our conditions are typically satisfied in case the Banach space is infinite dimensional.},
	keywords = {Gaussian process, High-resolution quantization, isoperimetric inequality., small ball probability},
	author = {Steffen Dereich and F. Fehringer and A. Matoussi and M. Scheutzow}
}
@article {GaussianCase,
	title = {Optimal quadratic quantization for numerics: the Gaussian case},
	journal = {Monte Carlo Methods and Applications},
	volume = {9},
	year = {2003},
	pages = {135{\textendash}166},
	abstract = {Optimal quantization has been recently revisited in multi-dimensional numerical integration (see [bib]SpaceQuantizationMethodIntegration[/bib]), multi-asset American option pricing (see [bib]stochasticNonLinearProblems[/bib]), control theory (see [bib]OptimalMarkovianQuantizationControl[/bib]) and nonlinear filtering theory (see [bib]PagesPhamNonLinearFiltering[/bib]). In this paper, we enlighten some numerical procedures in order to get some accurate optimal quadratic quantization of the Gaussian distribution in one and higher dimensions. We study in particular Newton method in the deterministic case (dimension $d = 1$) and stochastic gradient in higher dimensional case ($d \geq 2$). Some heuristics are provided which concern the step in the stochastic gradient method. Finally numerical examples borrowed from mathematical finance are used to test the accuracy of our Gaussian optimal quantizers.},
	keywords = {numerical integration, optimal quantization, stochastic gradient methods},
	attachments = {http://quantif.maths-fi.com/sites/default/files/Gaussian Case.pdf},
	author = {Gilles Pag{\`e}s and Jacques Printems}
}
@article {63,
	title = {Small ball probabilities around random centers of Gaussian measures and applications to quantization},
	journal = {J. Theor. Probab.},
	volume = {16(2)},
	year = {2003},
	pages = {427-449},
	abstract = {Let $\mu$ be a centered Gaussian measure on a separable Hilbert space $(E, \| \cdot \|)$. We are concerned with the logarithmic small ball probabilities around a $\mu$-distributed center $X$. It turns out that the asymptotic behavior of ${\textendash}\log \ \mu(B(X,\epsilon))$ is a.s. equivalent to that of a deterministic function $\phi_R (\epsilon)$. These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow [8].},
	keywords = {Gaussian process, quantization, small ball probabilites for random centers},
	author = {Steffen Dereich}
}
@article {GaussianProcessQuantization,
	title = {Functional quantization of Gaussian processes},
	journal = {Journal of Functional Analysis},
	volume = {196},
	number = {2},
	year = {2002},
	month = {December},
	pages = {486{\textendash}531},
	publisher = {Academic Press},
	abstract = {Quantization consists in studying the $L^r$ -error induced by the approximation of a random vector $X$ by a vector (quantized version) taking a finite number $n$ of values. For $R^m$-valued random vectors the theory and practice is quite well established and in particular, the asymptotics as $n\to \infty$ of the resulting minimal quantization error for nonsingular distributions is well known: it behaves like $c(X,r,m)n^{-1/m}$. This paper is a transposition of this problem to random vectors in an infinite dimensional Hilbert space and in particular, to stochastic processes $(X_t)_{t \in [0,1]}$ viewed as $L^2([0,1],dt)$-valued random vectors. For Gaussian vectors and the $L^2$-error we present detailed results for stationary and optimal quantizers. We further establish a precise link between the rate problem and Shannon{\textendash}Kolmogorov{\textquoteright}s entropy of $X$ : This allows us to compute the exact rate of convergence to zero of the minimal $L^2$-quantization error under rather general conditions on the eigenvalues of the covariance operator. Typical rates are $O(\log(n)^{-a}) , a > 0$. They are obtained, for instance, for the fractional Brownian motion and the fractional Ornstein{\textendash}Uhlenbeck process. The exponent a is closely related with the $L^2$-regularity of the process.
},
	keywords = {fractional Brownian motion, Gaussian process, quantization of probability distribution, Shannon{\textendash}Kolmogorov entropy, stationary processes},
	issn = {0022-1236},
	author = {Harald Luschgy and Gilles Pag{\`e}s}
}
@article {21,
	title = {Asymptotics of optimal quantizers for some scalar distributions},
	journal = {J. Comput. Appl. Math.},
	volume = {146},
	number = {2},
	year = {2002},
	pages = {253{\textendash}275},
	publisher = {Elsevier Science Publishers B. V.},
	address = {Amsterdam, The Netherlands, The Netherlands},
	abstract = {We obtain semi-closed forms for the optimal quantizers of some families of one-dimensional probability distributions. They yield the first examples of non-log-concave distributions for which uniqueness holds. We give two types of applications of these results. One is a fast computation of numerical approximations of one-dimensional optimal quantizers and their use in a multidimensional framework. The other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function, Laplace transform and characteristic function. Moreover, we obtain the rate of convergence in the Bucklew \& Wise Theorem and finally the asymptotic size of the Voronoi tessels.},
	issn = {0377-0427},
	author = {Fort, Jean-Claude and Gilles Pag{\`e}s}
}
@article {58,
	title = {The quantization dimension of self-similar probabilities},
	journal = {Math. Nachrichten},
	volume = {241},
	year = {2002},
	pages = {103-109 },
	author = {Harald Luschgy and Siegfried Graf}
}
@article {57,
	title = {Rates of convergence for the empirical quantization error},
	journal = {Ann. Probab.},
	volume = {30},
	year = {2002},
	pages = {874- 897},
	author = {Harald Luschgy and Siegfried Graf}
}
@article {59,
	title = {Asymptotics of the quantization errors for self-similar probabilities},
	journal = {Real Analysis Exchange 26},
	volume = {26(2)},
	year = {2001},
	pages = {795-810},
	author = {Harald Luschgy and Siegfried Graf}
}
@conference {4,
	title = {A quantization algorithm for solving multidimensional optimal stopping problems, preprint},
	booktitle = {University de Paris VI},
	year = {2001},
	pages = {1003{\textendash}1049},
	abstract = {A new grid method for computing the Snell envelope of a function of an $\mathbb{R}^d$-valued simulatable Markov chain $(X_k)_{0\leq k\leq n}$ is proposed. (This is a typical nonlinear problem that cannot be solved by the standard Monte Carlo method.) Every $X_k$ is replaced by a {\textquoteright}quantized approximation{\textquoteright} $\widehat{X}_k$  taking its values in a grid $\Gamma_k$ of size $N_k$. The $n$ grids and their transition probability matrices form a discrete tree on which a pseudo-Snell envelope is devised by mimicking the regular dynamic programming formula. Using the quantization theory of random vectors, we show the existence of a set of optimal grids, given the total number $N$ of elementary $\mathbb{R}^d$-valued quantizers. A recursive stochastic gradient algorithm, based on simulations of ($X_k)_{0\leq k\leq n}$, yields these optimal grids and their transition probability matrices. Some a priori error estimates based on the $L^p$-quantization errors $|X_k - \widehat{X}_k|^p$ are established. These results are applied to the computation of the Snell envelope of a diffusion approximated by its (Gaussian) Euler scheme. We apply these results to provide a discretization scheme for reflected backward stochastic differential equations. Finally, a numerical experiment is carried out on a two-dimensional American option pricing problem.},
	author = {Vlad Bally and Gilles Pag{\`e}s}
}
@article {stochasticNonLinearProblems,
	title = {A stochastic quantization method for nonlinear problems},
	journal = {Monte Carlo Methods and Applications},
	volume = {7(1)},
	year = {2001},
	pages = {21-34},
	abstract = {We give an algorithm based on the quantization method which computes the solution of multi-dimensional obstacle problems, e.g. the pricing of American option on a basket of assets. More generally this algorithm applies to semi-linear partial differential equations with free boundary, or, in probabilistic terms, to reflected backward stochastic differential equations. Some a priori estimates of the error are obtained. A numerical illustration about the pricing of American exchange options is proposed.},
	keywords = {algorithm, American exchange options, American option pricing, error bounds, free boundary, numerical examples, Optimal Stopping, quantization, Reflected Backward Stochastic Differential Equation, Snell envelope},
	author = {Vlad Bally and Gilles Pag{\`e}s and Jacques Printems}
}
@book {GrafLushgyMonograf,
	title = {Foundations of quantization for probability distributions},
	year = {2000},
	publisher = {Springer-Verlag New York, Inc.},
	organization = {Springer-Verlag New York, Inc.},
	address = {Secaucus, NJ, USA},
	abstract = {Due to the rapidly increasing need for methods of data compression, quantization has become a flourishing field in signal and image processing and information theory. The same techniques are also used in statistics (cluster analysis), pattern recognition, and operations research (optimal location of service centers). The book gives the first mathematically rigorous account of the fundamental theory underlying these applications. The emphasis is on the asymptotics of quantization errors for absolutely continuous and special classes of singular probabilities (surface measures, self-similar measures) presenting some new results for the first time. Written for researchers and graduate students in probability theory the monograph is of potential interest to all people working in the disciplines mentioned above.},
	keywords = {asymptotics, data compression, fractal geometry, information theory, multidimensional quantization, optimal quantizers, probability distributions},
	isbn = {3540673946},
	author = {Siegfried Graf and Harald Luschgy}
}
@article {SpaceQuantizationMethodIntegration,
	title = {A space quantization method for numerical integration},
	journal = {J. Comput. Appl. Math.},
	volume = {89},
	number = {1},
	year = {1998},
	pages = {1{\textendash}38},
	publisher = {Elsevier Science Publishers B. V.},
	address = {Amsterdam, The Netherlands},
	abstract = {We propose a new method (SQM) for numerical integration of $C^\alpha$  functions ($\alpha \in (0,2]$) defined on a convex subset $C$ of $\mathbb{R}^d$ with respect to a continuous distribution $\mu$. It relies on a space quantization of $C$ by a $n$-tuple $x:= (x_1,\cdots, x_n) \in C^n$. $\int f d\mu$ is approximated by a weighted sum of the $f (x_i)${\textquoteright}s. The integration error bound depends on the distortion $E_n^{z,\mu}(x)$ of the Vorono{\"\i} tessellation of $x$. This notion comes from Information Theoretists. Its main properties (existence of a minimizing $n$-tuple in $C^n$, asymptotics of $\min\limits_{C^n} E_n^{\alpha,\mu}$ as $n \to + \infty$) are presented for a wide class of measures $\mu$. A simple stochastic optimization procedure is proposed to compute, in any dimension $d$, $x^*$  and the characteristics of its Vorono{\"\i} tessellation. Some new results on the Competitive Learning Vector Quantization algorithm (when $\alpha = 2$) are obtained as a by-product. Some tests, simulations and provisional remarks are proposed as a conclusion.},
	keywords = {competitive algorithms, error estimation, learning algorithms, numerical integration, numerical methods, optimization method, vector quantization, Voronoi diagram},
	issn = {0377-0427},
	author = {Gilles Pag{\`e}s}
}
@article {60,
	title = {The quantization of the Cantor distribution},
	journal = {Math. Nachrichten},
	volume = {183},
	year = {1997},
	pages = {113-133},
	abstract = {For a real-valued random variable whose distribution is the classical Cantor probability, the $n$-quantization error and the $n$-optimal quantization rules are calculated for every natural number $n$. Moreover, the connection between the rate of convergence of the logarithms of the quantization errors for $n$ going to infinity and the Hausdorff dimension of the Cantor set is indicated.},
	keywords = {Cantor distribution, quantization},
	author = {Harald Luschgy and Siegfried Graf}
}
@book {81,
	title = {Vector quantization and signal compression},
	year = {1992},
	publisher = {Kluwer Academic Press},
	organization = {Kluwer Academic Press},
	author = {Allen Gersho and Robert M. Gray}
}
@article {16,
	title = {Asymptotic quantization error of continuous signals and the quantization dimension},
	journal = {IEEE Trans. Inform. Theory},
	volume = {IT-28},
	number = {2},
	year = {1982},
	month = {March},
	pages = {139{\textendash}149},
	abstract = {Extensions of the limiting quantization error formula of Bennet are proved. These are of the form $D_{s,k}(N,F)=N^{-\beta}B$, where $N$ is the number of output levels, $D_{s,k}(N,F)$ is the $s$th moment of the metric distance between quantizer input and output, $\beta, B \superior 0, k=s/\beta$ is the signal space dimension, and $F$ is the signal distribution. If a suitably well-behaved $k$-dimensional signal density $f(x)$ exists, $B=b_{s,k}[\int f^{rho}(x)dx]^{1/ rho},rho=k/(s+k)$, and $b_{s,k}$ does not depend on $f$. For $k=1,s=2$ this reduces to Bennett{\textquoteright}s formula. If $F$ is the Cantor distribution on $[0,1]$, $0\inferior k=s/ \beta= \log 2/ \log 3 \inferior 1$ and this $k$ equals the fractal dimension of the Cantor set $[12,13]$. Random quantization, optimal quantization in the presence of an output information constraint, and quantization noise in high dimensional spaces are also investigated.},
	keywords = {information-theory, nldr, rate-distortion, source-coding, vector-quantization},
	author = {Paul L. Zador}
}
@article {8,
	title = {Least squares quantization in PCM},
	journal = {Information Theory, IEEE Transactions on},
	volume = {28},
	number = {2},
	year = {1982},
	pages = {129{\textendash}137},
	abstract = {It has long been realized that in pulse-code modulation (PCM), with a given ensemble of signals to handle, the quantum values should be spaced more closely in the voltage regions where the signal amplitude is more likely to fall. It has been shown by Panter and Dite that, in the limit as the number of quanta becomes infinite, the asymptotic fractional density of quanta per unit voltage should vary as the one-third power of the probability density per unit voltage of signal amplitudes. In this paper the corresponding result for any finite number of quanta is derived; that is, necessary conditions are found that the quanta and associated quantization intervals of an optimum finite quantization scheme must satisfy. The optimization criterion used is that the average quantization noise power be a minimum. It is shown that the result obtained here goes over into the Panter and Dite result as the number of quanta become large. The optimum quantization schemes for $2^b$ quanta, $b=1,2,\cdots, 7$, are given numerically for Gaussian and for Laplacian distribution of signal amplitudes.},
	keywords = {clustering, k-means},
	author = {Stuart P. Lloyd}
}
@article {66,
	title = {Multidimensional asymptotic quantization theory with $r$th power distortion measures},
	journal = {Information Theory, IEEE Transactions on },
	volume = {28},
	year = {1982},
	pages = {239 - 247},
	abstract = {Asymptotic properties of therth power distortion measure associated with quantized $k$-dimensional random variables are considered. Subject only to a moment condition, it is shown that the infimum over all $N$ level quantizers of the quantity $N^{r/k}$ times the $r$th power average distortion converges to a finite constant as $N \rightarrow \infty$.},
	author = {James A. Bucklew and Gary L. Wise}
}
