Functional quantization of Gaussian processes

Asymptotically optimal quantization schemes for Gaussian processes on Hilbert spaces

“…For every n ∈ N, L 2 -optimal n-quantizers α ⊂ H exist, that is (E min a∈α X − a 2 ) 1/2 = e n (X) (see [10]). If card(supp(P X )) ≥ n, optimal n-quantizers α satisfy card(α) = n, P(X ∈ C a (α)) > 0 and the stationarity condition…”

“…Assume α to be an optimal n-quantizer [10] now states, that this orthonormal basis of U can be constructed by eigenvectors of C ζ , which correspond to the d * n largest eigenvalues. To be more precise, we get…”

“…However, when l = 1 (scalar product quantizers made up with blocks of fixed length l = 1, see Design IV), one checks that these estimates still stand as equalities since orthogonality can now be substituted by the independence of ξ j −ξ j and stationarity property (2.2) of the quantizationsξ j , j ≥ 1. It is often good enough for applications to use scalar product quantizers (see [10,17]). If, for instance δ = 1, then…”

Entropy-constrained functional quantization of Gaussian processes

“…See Luschgy and Pagès [10], [11]. Recently, the high-rate behaviour of quantization when the rate is measured by the entropy of the quantizers has attracted new attention in finite dimensions.…”

A Rbsdes Approach for the Numerical Evaluation of the American Option Pricing Problem

“…Quantization methods have been also extensively investigated in Information Theory and signal processing [22], while, since the early 2000's, the associated infinite dimensional setting has been investigated from both theoretical and numerical point of view, with a special attention paid to functional quantization, see [26,29]. Other types of quantization method, as Marginal quantization and Markovian quantization, were proposed in [28], the first one, relies on an optimal approximation of the marginal distributions of the process, while the second one, looks for an optimal approximation of transition probabilities of the Markov process at some points.…”