Intermediate rank lattice rules and applications to finance
type: Item
instance of: scholarly article
title: Intermediate rank lattice rules and applications to finance
zbMATH Open document ID: 1161.65003
author: Yongzeng Lai
published in: Applied Numerical Mathematics
publication date: 2008-12-16T00:00:00Z
review text: In scientific computation the Monte Carlo (MC) simulation method is the main tool that deals with high-dimensional problems. The quasi-Monte Carlo (QMC) method uses quasi-random or low-discrepancy point sets to reduce the rate of convergence of the obtained errors. The author discusses the intermediate rank lattice rule for the general case when the number of quadrature points is \(n^{t}.m,\) where \(n,m\) are integers such that \((n,m)=1\) and \(t\) is the rank of the rule. In Section 2 the idea of the lattice rule for the calculus of approximations of high-dimensional integrals of one-periodic in each argument integrands is developed. Some known results of the expression of the lattice rule are reminded. Notions as a good lattice rule, a maximum rank rule and intermediate rank lattice rules are discussed. The functional class \(S_{s}^{\alpha}(C)\) is considered. The lattice rule error is bounded in terms of the quantity \(P_{\alpha}({\mathbf g},N).\) The quantity \(M_{\alpha}(N)\) which is the mean of \(P_{\alpha}({\mathbf g},N)\) over the generating vector \({\mathbf g}\) is introduced. Theorem 2 gives an expression of \(M_{\alpha}(N)\) in the terms of the Riemann zeta function and the order \( {\mathcal O}\left({(\log N)^{\alpha s} \over N^{\alpha}}\right)\) of \((M_{\beta(N)}(N))^{\alpha /\beta(N)},\) for a special value of the parameter \(\beta(N)\) is obtained. In Section 3 intermediate rank lattice rules are considered. The existence of good rank \(t\) rules is established. The mean \(M_{\alpha,t}^{(n)}(m)\) of \(P_{\alpha}(Q_{t})\) over a subset of \(\mathbb{Z}^{s}\) is introduced. Theorem 3 is the main theoretical result of the paper and gives an expression of \(M_{\alpha,t}^{(n)}(m).\) Under the conditions of Theorem 3 in Corollary 1 an upper bound of \(M_{\alpha,t}^{(n)}(m)\) is given and the order \({\mathcal O}\left( {\log \log m \over m}\right)\) as \(m \to \infty\) of \(M_{\alpha,t}^{(n)}(m)\) is obtained. Theorem 4 states the existence of good inter…
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 11K45
Mathematics Subject Classification ID: 91G60
zbMATH DE Number: 5382926
zbMATH Keywords: high dimensional numerical integration
zbMATH Keywords: intermediate rank lattice rules
zbMATH Keywords: computational finance
zbMATH Keywords: Monte Carlo method
zbMATH Keywords: quasi-Monte Carlo method
zbMATH Keywords: error bound
zbMATH Keywords: low-discrepancy point sets
zbMATH Keywords: convergence
zbMATH Keywords: quadrature
zbMATH Keywords: good lattice rule
zbMATH Keywords: maximum rank rule
zbMATH Keywords: Riemann zeta function
zbMATH Keywords: Korbov type lattice rules
zbMATH Keywords: option pricing
zbMATH Keywords: Sobol' sequences
describes a project that uses: QSIMVN
reviewed by: Vassil Grozdanov
MaRDI profile type: Publication
full work available at URL: j.apnum.2007.11.024
OpenAlex ID: W2078698791
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DOI: 10.1016/J.APNUM.2007.11.024
description: scientific article; zbMATH DE number 5382926