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2014 | Buch

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Introduction to Quasi-Monte Carlo Integration and Applications

verfasst von: Gunther Leobacher, Friedrich Pillichshammer

Verlag: Springer International Publishing

Buchreihe : Compact Textbooks in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook introduces readers to the basic concepts of quasi-Monte Carlo methods for numerical integration and to the theory behind them. The comprehensive treatment of the subject with detailed explanations comprises, for example, lattice rules, digital nets and sequences and discrepancy theory. It also presents methods currently used in research and discusses practical applications with an emphasis on finance-related problems. Each chapter closes with suggestions for further reading and with exercises which help students to arrive at a deeper understanding of the material presented.

The book is based on a one-semester, two-hour undergraduate course and is well-suited for readers with a basic grasp of algebra, calculus, linear algebra and basic probability theory. It provides an accessible introduction for undergraduate students in mathematics or computer science.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

In this book we consider the problem of numerical integration over the s-dimensional unit cube [0, 1]^s, $\displaystyle{ \int _{[0,1]^{s}}f(\boldsymbol{x})\,\mathrm{d}\boldsymbol{x} =\int _{ 0}^{1}\cdots \int _{ 0}^{1}f(x_{ 1},\ldots,x_{s})\,\mathrm{d}x_{1}\ldots \,\mathrm{d}x_{s}. }$ Here the dimension s may be large in practical applications. The restriction to integration problems over the unit cube [0, 1]^s is mostly for simplicity and in many cases does not impose a big limitation, since most integrals over bounded or unbounded regions can be transformed into integrals over the unit cube (although one has to be careful in choosing suitable transformations which, of course, have influence on the behavior of the transformed integrand).

Gunther Leobacher, Friedrich Pillichshammer

2. Uniform Distribution Modulo One

Abstract

The theory of Uniform Distribution Modulo One is a branch of Number Theory which goes back to the seminal work of H. Weyl from 1916. For us the main motivation to study this topic lies in its application for numerical integration based on QMC rules.

Gunther Leobacher, Friedrich Pillichshammer

3. QMC Integration in Reproducing Kernel Hilbert Spaces

Abstract

We return to the problem of numerical integration of multivariate functions. As already mentioned in Sect. 1.1, we normalize the integration domain to be the compact unit cube [0, 1]^s, and hence the integrals considered are of the form (1.1).

Gunther Leobacher, Friedrich Pillichshammer

4. Lattice Point Sets

Abstract

We have shown in Proposition 2.6 that the infinite sequence $(\{n\boldsymbol{\alpha }\})_{n\in \mathbb{N}_{0}}$ is uniformly distributed modulo one under a certain condition on the vector $\boldsymbol{\alpha }\in \mathbb{R}^{s}$. In this chapter we consider “finite” versions of such sequences which are referred to as lattice point sets.

Gunther Leobacher, Friedrich Pillichshammer

5. (t, m, s)-Nets and (t, s)-Sequences

Abstract

We are interested in point sets with very low star discrepancy. This means that we aim on finding point sets $\mathcal{P}$ for which the absolute local discrepancy,

$$\displaystyle{\vert \varDelta _{\mathcal{P},N}(\boldsymbol{y})\vert = \left \vert \frac{A([\mathbf{0},\boldsymbol{y}),\mathcal{P},N)} {N} -\lambda _{s}([\mathbf{0},\boldsymbol{y}))\right \vert,}$$

is as small as possible for all $\boldsymbol{y} \in (0,1]^{s}$.

Gunther Leobacher, Friedrich Pillichshammer

6. A Brief Discussion of the Discrepancy Bounds

Abstract

In many applications the dimension s can be rather large. In this case, the asymptotically almost optimal bounds on the discrepancy which we obtained, e.g., for the Hammersley point set or for (t, m, s)-nets soon become useless for a modest number N of points. For example, assume that for every $s,N \in \mathbb{N}$ we have a point set $\mathcal{P}_{s,N}$ in the s-dimensional unit cube of cardinality N with star discrepancy of at most

$$\displaystyle\begin{array}{rcl} D_{N}^{{\ast}}(\mathcal{P}_{ s,N}) \leq c_{s}\frac{(\log N)^{s-1}} {N},& &{}\end{array}$$

(6.1)

with some c _s > 0 that is independent of N.

Gunther Leobacher, Friedrich Pillichshammer

7. Basics of Financial Mathematics

Abstract

In this chapter we will give some background on mathematical finance, or, to be precise, on the mathematical theory that lies behind derivative pricing. Since the 1980s, financial mathematics has become a huge field that uses methods from many other branches of mathematics, most notably from probability theory. The reliance on probability theory provides us with a wealth of applications for simulation techniques.

Gunther Leobacher, Friedrich Pillichshammer

8. Monte Carlo and Quasi-Monte Carlo Simulation

Abstract

Is this chapter we will learn the basics of pricing derivatives using simulation methods. We will consider both Monte-Carlo and quasi-Monte Carlo but – of course – with a special emphasis on the latter. The aim of our exposition is not to provide a large toolbox for the quantitative analyst, but to help getting started with the topic. QMC-pricing is an active area of research by its own and the reader is encouraged to consult the specialized literature. We will, however, take a look at some popular examples that frequently serve as benchmarks for refined simulation techniques.

Gunther Leobacher, Friedrich Pillichshammer

Backmatter

Titel: Introduction to Quasi-Monte Carlo Integration and Applications
verfasst von: Gunther Leobacher
Friedrich Pillichshammer
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-03425-6
Print ISBN: 978-3-319-03424-9
DOI: https://doi.org/10.1007/978-3-319-03425-6

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Uniform Distribution Modulo One

3. QMC Integration in Reproducing Kernel Hilbert Spaces

4. Lattice Point Sets

5. (t, m, s)-Nets and (t, s)-Sequences

6. A Brief Discussion of the Discrepancy Bounds

7. Basics of Financial Mathematics

8. Monte Carlo and Quasi-Monte Carlo Simulation

Backmatter

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