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E-raamat: Sparse Grid Quadrature in High Dimensions with Applications in Finance and Insurance

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Sparse Grid Quadrature in High Dimensions with Applications in Finance and InsuranceSuurem pilt
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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.
1 Introduction
1(10)
2 Dimension-wise Decompositions
11(18)
2.1 Classical ANOVA Decomposition
13(8)
2.1.1 Effective Dimensions
16(3)
2.1.2 Error Bounds
19(2)
2.2 Anchored-ANOVA Decomposition
21(8)
2.2.1 Effective Dimensions
22(4)
2.2.2 Error Bounds
26(3)
3 Dimension-wise Quadrature
29(22)
3.1 Classical Multivariate Quadrature Methods
29(14)
3.1.1 Monte Carlo
31(3)
3.1.2 Quasi-Monte Carlo
34(4)
3.1.3 Product Methods
38(5)
3.2 Dimension-wise Quadrature Methods
43(8)
3.2.1 Truncation and Discretization
43(2)
3.2.2 Error and Costs
45(2)
3.2.3 A priori Construction
47(2)
3.2.4 Dimension-adaptive Construction
49(2)
4 Sparse Grid Quadrature
51(26)
4.1 Sparse Grid Methods
51(12)
4.1.1 Classical Construction
52(3)
4.1.2 Delayed Basis Sequences
55(4)
4.1.3 Generalised Sparse Grids
59(2)
4.1.4 Dimension-adaptive Sparse Grids
61(2)
4.2 Optimal Sparse Grids in Weighted Spaces
63(11)
4.2.1 Cost-Benefit Ratio
64(3)
4.2.2 Cost Analysis
67(2)
4.2.3 Error Analysis
69(2)
4.2.4 Analysis of Error versus Cost
71(3)
4.3 Relation to Dimension-wise Quadrature
74(3)
5 Dimension Reduction and Smoothing
77(24)
5.1 Dimension Reduction
77(12)
5.1.1 Random Walk, Brownian Bridge, PCA
78(4)
5.1.2 Linear transformations
82(7)
5.2 Domain Decomposition
89(12)
5.2.1 Root Finding
89(1)
5.2.2 Hyperplane Arrangements
90(8)
5.2.3 Conditional Sampling
98(3)
6 Validation and Applications
101(52)
6.1 Interest Rates Derivatives
104(10)
6.1.1 Zero Coupon Bonds
104(5)
6.1.2 Collateralized Mortgage Obligations
109(5)
6.2 Path-dependent Options
114(11)
6.2.1 Asian options
115(5)
6.2.2 Barrier options
120(5)
6.3 Performance-dependent Options
125(11)
6.3.1 Framework and Pricing Formulas
125(6)
6.3.2 Numerical Results
131(5)
6.4 Asset-Liability Management in Life Insurance
136(13)
6.4.1 Model and Integral Representation
136(9)
6.4.2 Numerical Results
145(4)
6.5 Summary and Discussion
149(4)
7 Summary and Conclusions
153(4)
A Discrepancy and Tractability
157(22)
A.1 Reproducing Kernel Hilbert Spaces
157(4)
A.2 Notions of Discrepancy
161(6)
A.3 Tractability Results
167(12)
References
172(7)
Index 179
Scientific employee at the Institute for Numerical Simulation at the University of Bonn (July 2004 - January 2009) Involved in several teaching activities and research projects in the area of computational finance partly in close cooperation with financial institutions Since January 2009 at head office of Baloise Group working on the introduction of stochastic models for life insurance portfolios
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