Research Article | Open Access
Stefan Junglen, Harald Luschgy, "A Constructive Sharp Approach to Functional Quantization of Stochastic Processes", Journal of Applied Mathematics, vol. 2010, Article ID 378519, 32 pages, 2010. https://doi.org/10.1155/2010/378519
A Constructive Sharp Approach to Functional Quantization of Stochastic Processes
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, Lévy processes.
We consider a separable Banach space and a Borel random variable : with finite th moment for some .
For a given natural number , the quantization problem consists in finding a set that minimizes over all subsets with card . Such sets are called -codebooks or -quantizers. The corresponding infimum is called the th -quantization error of in , and any -quantizer fulfilling is called -optimal -quantizer. For a given -quantizer one defines the nearest neighbor projection where the Voronoi partition is defined as a Borel partition of satisfying The random variable is called -quantization of . One can easily verify that is the best quantization of in , which means that for every random variable with values in we have
Applications of quantization go back to the 1940s, where quantization was used for the finite-dimensional setting , called optimal vector quantization, in signal compression and information processing (see, e.g., [1, 2]). Since the beginning of the 21st century, quantization has been applied for example in finance, especially for pricing path-dependent and American style options. Here, vector quantization  as well as functional quantization [4, 5] is useful. The terminology functional quantization is used when the Banach space is a function space, such as or . In this case, the realizations of can be seen as the paths of a stochastic process.
A question of theoretical as well as practical interest is the issue of high-resolution quantization which concerns the behavior of when tends to infinity. By separability of , we can choose a dense subset and we can deduce in view of that tends to zero as tends to infinity.
A natural question is then if it is possible to describe the asymptotic behavior of . It will be convenient to write for sequences and if , if and if .
In the finite-dimensional setting this behavior can satisfactory be described by the Zador Theorem (see ) for nonsingular distributions . In the infinite dimensional case, no such global result holds so far, without some additional restrictions. To describe one of the most famous results in this field, we call a measurable function for an regularly varying at infinity with index if for every
Theorem 1.1 (see ). Let be a centered Gaussian random variable with values in the separable Hilbert space and the decreasing eigenvalues of the covariance operator :, (which is a symmetric trace class operator). Assume that for some regularly varying function with index . Then, the asymptotics of the quantization error is given by where .
Note that any change of ~ in the assumption that to either ≲, ≈ or ≳ leads to the same change in (1.9). Theorem 1.1 can also be extended to an index (see ). Furthermore, a generalization to an arbitrary moment (see ) as well as similar results for special Gaussian random variables and diffusions in non-Hilbertian function spaces (see, e.g., [9–11]) have been developed. Moreover, several authors established a precise link between the quantization error and the behavior of the small ball function of a Gaussian measure (see, e.g., [12, 13]) which can be used to derive asymptotics of quantization errors. More recently, for several types of Lèvy processes (sharp) optimal rates have been developed by several authors (see, e.g., [14–17]).
Coming back to the practical use of quantizers as a good approximation for a stochastic process, one is strongly interested in a constructive approach that allows to implement the coding strategy and to compute (at least numerically) good codebooks.
Considering again Gaussian random variables in a Hilbert space setting, the proof of Theorem 1.1 shows us how to construct asymptotically -optimal -quantizers for these processes, which means that sequences of -quantizers satisfy These quantizers can be constructed by reducing the quantization problem to a quantization problem of a finite-dimensional normal distributed random variable. Even if there are almost no explicit formulas known for optimal codebooks in finite dimensions, the existence is guaranteed (see [6, Theorem 4.12]) and there exist a lot of deterministic and stochastic numerical algorithms to compute optimal codebooks (see e.g., [18, 19] or ). Unfortunately, one needs to know explicitly the eigenvalues and eigenvectors of the covariance operator to pursue this approach.
If we consider other non-Hilbertian function spaces or non-Gaussian random variables in an infinite-dimensional Hilbert space, there is much less known on how to construct asymptotically optimal quantizers. Most approaches to calculate the asymptotics of the quantization error are either non-constructive (e.g., [12, 13]) or tailored to one specific process type (e.g., [9–11]) or the constructed quantizers do not achieve the sharp rate in the sense of (1.10) (e.g.,  or ) but just the weak rate
In this paper, we develop a constructive approach to calculate sequences of asymptotically -optimal -quantizers (in the sense of (1.10)) for a broad class of random variables in infinite dimensional Banach spaces (Section 2). Constructive means in this case that we reduce the quantization problem to the quantization problem of a -valued random variable, that can be solved numerically. This approach can either be used in Hilbert spaces in case the eigenvalues and eigenvectors of the covariance operator of a Gaussian random variable are unknown (Sections 3.1 and 3.2), or for quantization problems in different Banach spaces (Sections 4 and 5).
In Section 4, we discuss Gaussian random variables in . This part is related to the PhD thesis of Wilbertz . More precisely, we sharpen his constructive results by showing that the quantizers constructed in the thesis also achieve the sharp rate for the asymptotic quantization error (in the sense of (1.10)). Moreover, we can show that the dimensions of the subspaces wherein these quantizers are contained can be lessened without loosing the sharp asymptotics property.
It is worth mentioning that all these quantizers can be constructed without knowing the true rate of the quantization error. This means precisely that we know a (rough) lower bound for the quantization error, that is, and the true but unknown rate is for constants , then, we are able to construct a sequence of -quantizers , that satisfies for the optimal but still unknown constants .
The crucial factors for the numerical implementation are the dimensions of the subspaces, wherein the asymptotically optimal quantizers are contained. We will calculate the dimensions of the subspaces obtained through our approach, and we will see that for all analyzed Gaussian processes, and also for many Lévy processes we are very close to the known asymptotics of the optimal dimension in the case of Gaussian processes in infinite-dimensional Hilbert spaces.
Notations and Definitions
If not explicitly differently defined, the following notations hold throughout the paper. (i)We denote by a Borel random variable in the separable Banach space with . (ii) will always denote the norm in whereas will denote the norm in . (iii)The scalar product in a Hilbert space will be denoted by . (iv)The smallest integer above a given real number will be denoted by . (v)A sequence is called admissible for a centered Gaussian random variable in if and only if for any sequence of independent -distributed random variables it holds that converges . in and . An admissible sequence is called rate optimal for in if and only if as . A precise characterization of admissible sequences can be found in . (vi)An orthonormal system (ONS) is called rate optimal for in the Hilbert space if and only if as .
2. Asymptotically Optimal Quantizers
The main idea is contained in the subsequent abstract result. The proof is based on the following elementary but very useful properties of quantization errors.
Lemma 2.1 (see ). Let , be separable Banach spaces, a random variable in E, and . (1)If is Lipschitz continuous with Lipschitz constant , then and for every -quantizer for it holds that (2)Let be linear, surjective, and isometric. Then, for and and for every -quantizer for it holds that
To formulate the main result, we need for an infinite subset the following.
Condition 1. There exist linear operators for with , for finite dimensional subspaces with , where the norm is defined as
Condition 2. There exist linear isometric and surjective operators with suitable norms in for all .
Condition 3. There exist random variables for in with , such that for the approximation error it holds that as along .
Remark 2.2. The crucial point in Condition 1 is the norm one restriction for the operators . Condition 2 becomes Important when constructing the quantizers in equipped with, in the best case, some well-known norm. As we will see in the proof of the subsequent theorem, to show asymptotic optimality of a constructed sequence of quantizers one needs to know only a rough lower bound for the asymptotic quantization error. In fact, this lower bound allows us in combination with Condition 3 to choose explicitly a sequence , such that
Theorem 2.3. Assume that Conditions 1–3 hold for some infinite subset . One chooses a sequence such that (2.7) is satisfied. For , let be an -optimal -quantizer for in .
Then, is an asymptotically -optimal sequence of -quantizers for in and as .
Remark 2.4. Note, that for there always exist -optimal -quantizers for ([6, Theorem 4.12]).
Proof. Using Condition 3 and the fact that for all since , we can choose a sequence fulfilling (2.7). Using Lemma 2.1 and Condition 2, we see that is an -optimal -quantizer for in . Then, by using Condition 1, (2.7), and Lemma 2.1 we get The last equivalence of the assertion follows from (1.6).
Remark 2.5. We will usually choose for all , with an exception in Section 3 and .
Remark 2.6. The crucial factor for the numerical implementation of the procedure is the dimensions of the subspaces . For the well-known case of the Brownian motion in the Hilbert space it is known that this dimension sequence can be chosen as , . In the following examples we will see that we can often obtain similar orders like for constants just slightly higher than one.
We point out that there is a nonasymptotic version of Theorem 2.3 for nearly optimal -quantizers, that is, for -quantizers, which are optimal up to . Its proof is analogous to the proof of Theorem 2.3.
3. Gaussian Processes with Hilbertian Path Space
In this chapter, let be a centered Gaussian random variable in the separable Hilbert space . Following the approach used in the proof of Theorem 1.1, we have for every sequence of independent -distributed random variables where denote the eigenvalues and denote the corresponding orthonormal eigenvectors of the covariance operator of (Karhunen-Loève expansion). If these parameters are known, we can choose a sequence such that a sequence of optimal quantizer for is asymptotically optimal for in .
In order to construct asymptotically optimal quantizers for Gaussian random variables with unknown eigenvalues or eigenvectors of the covariance operator, we start with more general expansions. In fact, we just need one of the two orthogonalities, either in or in .
Before we will use these representations for to find suitable triples as in Theorem 2.3, note that for Gaussian random variables in fulfilling suitable assumptions we know that(1)Let be an orthonormal basis of . Then Compared to (3.1) we see that are still Gaussian but generally not independent. (2)Let be an admissible sequence for in such that Compared to (3.1) the sequence is generally not orthogonal. for all ; see . Thus, we will focus on the case to search for lower bounds for the quantization errors.
3.1. Orthonormal Basis
Let be an orthonormal basis of . For the subsequent subsection we use the following notations. (1)We set .(2) We set , the orthogonal projection on . It is well known that . (3)Define the linear, surjective, and isometric operators by where denotes the th unit vector in , .
Theorem 3.1. Assume that the eigenvalue sequence of the covariance operator satisfies for , and let be arbitrary. Assume further that is a rate optimal ONS for in . One sets for . Then, one gets for every sequence of -optimal -quantizers for in the asymptotics as .
Proof. Let be the corresponding orthonormal eigenvector sequence of . Classic eigenvalue theory yields for every Combining this with rate optimality of the ONS for , we get Using the equivalence of the -norms of Gaussian random variables ([23, Corollary 3.2]), and since is Gaussian, we get for all With as in Theorem 1.1, we get by using (3.4) and Theorem 1.1 the weak asymptotics . Therefore, the sequence satisfies (2.7) since and the assertion follows from Theorem 2.3.
3.2. Admissible Sequences
In order to show that linear operators similar to those used in the subsection above are suitable for the requirements of Theorem 2.3, we need to do some preparations. Since the covariance operator of a Gaussian random variable is symmetric and compact (in fact trace class), we will use a well-known result concerning these operators. This result can be used for quantization in the following way.
Lemma 3.2. Let be a centered Gaussian random variable with values in the Hilbert space and , where and are independent centered Gaussians. Then Let , be the positive monotone decreasing eigenvalues of , and . Then, for it holds that
Proof. Since are independent centered Gaussians, we have for all . This easily leads to The covariance operator of a centered Gaussian random variable is positive semidefinite. Hence, by using a result on the relation of the eigenvalues of those operators (see, e.g., [24, page 213]), we get inequalities (3.12).
Let be an admissible sequence for , and assume that a.s. In this subsection, we use the following notations.(1)We set . (2) We define by for and for , where and denote the eigenvalues and the corresponding eigenvectors of and and the eigenvalues and the corresponding eigenvectors of , with defined as Note that maps onto since Furthermore, it is important to mention that one does not need to know and explicitly to construct the subsequent quantizers, since we can find for any a random variable such that (see the proof of Theorem 3.3), which is explicitly known and sufficient to know for the construction.(3)Define the linear, surjective, and isometric operators by where denotes the th unit vector of for .
Theorem 3.3. Assume that the eigenvalue sequence of the covariance operator satisfies for , and let arbitrary. Assume that is a rate optimal admissible sequence for in . One sets for . Then, there exist random variables , with such that for every sequence of -optimal -quantizers for in as .
Proof. Linearity of follows from the orthogonality of the eigenvectors. In view of the inequalities for the eigenvalues in Lemma 3.2 and the orthonormality of the family , we have for every with
such that .
Note next that for every there exist independent -distributed random variables satisfying Then, we choose random variables such that is a sequence of independent -distributed random variables. We set and get by using rate optimality of the admissible sequences and where rate optimality of is a consequence of Using the equivalence of the -norms of Gaussian random variables ([23, Corollary 3.2]), and since is Gaussian, we get for all With as in Theorem 1.1, we get by using (3.4) and Theorem 1.1 the weak asymptotics , . Therefore, the sequence satisfies (2.7) since and the assertion follows from Theorem 2.3.
3.3. Comparison of the Different Schemes
At least in the case , we have a strong preference for using the method as described in Section 3.1. We use the notations as in the above subsections including an additional indexation for and , where , for , are defined as in Theorems 3.1 and 3.3. Note that for this purpose the size of the codebook and the size of the subspaces can be chosen arbitrarily (i.e., does not depend on ). The ONS is chosen as the ONS derived with the Gram-Schmidt procedure from the admissible sequence for the Gaussian random variable in the Hilbert space , such that the definition of coincides in the two subsections.
Proposition 3.4. It holds for that
Proof. Consider for the decomposition . The key is the orthogonality of to , , and , which gives the two equalities in the following calculation: The inequality (*) follows from the optimality of the codebook for .
4. Gaussian Processes with Paths in
In the previous section, where we worked with Gaussian random variables in Hilbert spaces, we saw that special Hilbertian subspaces, projections, and other operators linked to the Gaussian random variable were good tools to develop asymptotically optimal quantizers based on Theorem 2.3. Since we now consider the non-Hilbertian separable Banach space , we have to find different tools that are suitable to use Theorem 2.3.
The tools used in  are B-splines of order . In the case , that we will consider in the sequel, these splines span the same subspace of as the classical Schauder basis. We set for , , and the knots and the hat functions For the remainder of this subsection, we will use the following notations.(1)As subspaces we set .(2) As linear and continuous operators we set the quasiinterpolant where .(3) The linear and surjective isometric mappings one defines as It is easy to see that holds for every .
For the application of Theorem 2.3, we need to know the error bounds for the approximation of with the quasiinterpolant . For Gaussian random variables, we can provide the following result based on the smoothness of an admissible sequence for in .
Proposition 4.1. Let be admissible for the centered Gaussian random variable in . Assume that (1) for every , and , (2) with for every and . Then, for any and some constant it holds that for every .
Proof. Using of [25, Theorem 1], we get for an arbitrary , some constant , and every . Thus, we have Using of [26, Chapter 7, Theorem 7.3], we get for some constant where the module of smoothness is defined by For an arbitrary we have by using Taylor expansion Combining this, we get for an arbitrary and constants , using again the equivalence of Gaussian moments, To minimize over k, we choose . Thus, we get for some constant and an arbitrary
Now, we are able to prove the main result of this section.
Theorem 4.2. Let be a centered Gaussian random variable and an admissible sequence for in fulfilling the assumptions of Proposition 4.1 with , where the constant satisfies with denoting the monotone decreasing eigenvalues of the covariance operator of in and . One sets for some . Then, for every sequence of -optimal -quantizers for in , it holds that as .
Proof. For every , with it holds that
since are partitions of the one for every , so that .
We get a lower bound for the quantization error from the inequality for all . Consequently, we have From Theorem 1.1 and (3.4) we obtain where is given as in Theorem 1.1. Finally, we get by combining (4.16) and Proposition 4.1 for sufficiently small and the assertion follows from Theorem 2.3.
5. Processes with Path Space
Another useful tool for our purposes is the Haar basis in for , which is defined by This is an orthonormal basis of and a Schauder basis of for , that is, converges to in for every ; see .
The Haar basis was used in  to construct rate optimal sequences of quantizers for mean regular processes. These processes are specified through the property that for all where is regularly varying with index at 0, which means that for all . Condition (5.2) also guarantees that the paths lie in .
For our approach, it will be convenient to define for and the knots and for the functions and the operators Note that for , , and
For the remainder of the subsection, we set the following. (1)We set for the subspaces .(2) Set the linear and continuous operator to (3)For we set the isometric isomorphisms as
Theorem 5.1. Let be a random variable in the Banach space for some fulfilling the mean pathwise regularity property for constants and . Moreover, assume that for constants . Then, for an arbitrary and it holds that every sequence of -optimal -quantizers for in satisfies as .
Proof. As in the above subsections, we check that the sequences and satisfy Conditions 1–3. Since , where is defined by we get for , with and by using Jensen's inequality, and thus . The operators satisfy Condition 2 of Theorem 2.3 since For Condition 3, we note that for