Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance
This book deals with the numerical analysis and efficient numerical treatment of high-dimensional integrals using sparse grids and other dimension-wise integration techniques with applications to finance and insurance. The book focuses on providing insights into the interplay between coordinate transformations, effective dimensions and the convergence behaviour of sparse grid methods. The techniques, derivations and algorithms are illustrated by many examples, figures and code segments. Numerical experiments with applications from finance and insurance show that the approaches presented in this book can be faster and more accurate than (quasi-) Monte Carlo methods, even for integrands with hundreds of dimensions.
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Sparse Grid Quadrature in High Dimensions with Applications in Finance and ...
No preview available - 2012
Algorithm ALM model anchored-ANOVA decomposition anchored-ANOVA terms ANOVA approach approximation Asian options asset barrier options bonus Brownian bridge Brownian motion compute convergence behaviour convergence rate corresponding d-dimensional defined denotes dimension reduction dimension-adaptive sparse grid dimension-wise quadrature methods discretization error domain decomposition efficient error bound Example formulas function f Gauss-Hermite Gaussian weight high dimensions high-dimensional integrals Hilbert space hyperplane arrangement index set integrand kernel Hilbert space L2-star discrepancy lattice rules Lemma low effective dimension LT-construction matrix Monte Carlo methods multivariate nominal dimension normally distributed numerical methods numerical quadrature obtain optimal orthant path construction payoff performance performance-dependent options point sets polynomial QMC methods quadrature rules quasi-Monte Carlo methods reduce reproducing kernel Hilbert sampling Section sequence SG methods simulation smooth Sobolev space sparse grid methods superposition dimension tion tractability transformation truncation dimension unit cube values variance vector worst case error