A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

Contents

Preface

Symbols

1   Prologue

1.1     About This Book

1.2     About the Concepts

1.3     About the Notation

1.4     Riemann, Stieltjes, and Burkill Integrals

1.5     The -Complete Integrals

1.6     Riemann Sums in Statistical Calculation

1.7     Random Variability

1.8     Contingent and Elementary Forms

1.9     Comparison With Axiomatic Theory

1.10   What Is Probability?

1.11   Joint Variability

1.12   Independence

1.13   Stochastic Processes

2   Introduction

2.1    Riemann Sums in Integration

2.2    The -Complete Integrals in Domain ]0,1]

2.3    Divisibility of the Domain ]0, 1]

2.4    Fundamental Theorem of Calculus

2.5    What Is Integrability?

2.6    Riemann Sums and Random Variability

2.7    How to Integrate a Function

2.8    Extension of the Lebesgue Integral

2.9    Riemann Sums in Basic Probability

2.10  Variation and Outer Measure

2.11  Outer Measure and Variation in [0, 1]

2.12  The Henstock Lemma

2.13  Unbounded Sample Spaces

2.14  Cauchy Extension of the Riemann Integral

2.15  Integrability on ]0, ∞[

2.16  “Negative Probability”

2.17  Henstock Integration in Rⁿ

2.18  Conclusion

3   Infinite-Dimensional Integration

3.1    Elements of Infinite-Dimensional Domain

3.2    Partitions of R^T

3.3    Regular Partitions of R^T

3.4    δ-Fine Partially Regular Partitions

3.5    Binary Partitions of R^T

3.6    Riemann Sums in R^T

3.7    Integrands in R^T

3.8    Definition of the Integral in R^T

3.9    Integrating Functions in R^T

4   Theory of the Integral

4.1    The Henstock Integral ...