Patrick Muldowney - Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics

1 Cover

2 Title Page Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics Patrick Muldowney

3 Copyright

4 Preface

5 Reading this BooknotesSet Note

6 Introduction Notes

7 Part I: Stochastic Calculus Chapter 1: Stochastic Integration Notes Chapter 2: Random Variation 2.1 What is Random Variation? 2.2 Probability and Riemann Sums 2.3 A Basic Stochastic Integral 2.4 Choosing a Sample Space 2.5 More on Basic Stochastic Integral Notes Chapter 3: Integration and Probability 3.1 ‐Complete Integration 3.2 Burkill‐complete Stochastic Integral 3.3 The Henstock Integral 3.4 Riemann Approach to Random Variation 3.5 Riemann Approach to Stochastic Integrals Notes Chapter 4: Stochastic Processes 4.1 From to 4.2 Sample Space with Uncountable 4.3 Stochastic Integrals for Example 12 4.4 Example 12 4.5 Review of Integrability Issues Notes Chapter 5: Brownian Motion5.1 Introduction to Brownian Motion 5.2 Brownian Motion Preliminaries 5.3 Review of Brownian Probability 5.4 Brownian Stochastic Integration 5.5 Some Features of Brownian Motion 5.6 Varieties of Stochastic Integral Notes Chapter 6: Stochastic Sums 6.1 Review of Random Variability 6.2 Riemann Sums for Stochastic Integrals 6.3 Stochastic Sum as Observable 6.4 Stochastic Sum as Random Variable 6.5 Introduction to 6.6 Isometry Preliminaries 6.7 Isometry Property for Stochastic Sums 6.8 Other Stochastic Sums 6.9 Introduction to Itô’s Formula 6.10 Itô’s Formula for Stochastic Sums 6.11 Proof of Itô’s Formula 6.12 Stochastic Sums or Stochastic Integrals? Notes

8 Part II: Field Theory Chapter 7: Gauges for Product Spaces7.1 Introduction 7.2 Three‐dimensional Brownian Motion 7.3 A Structured Cartesian Product Space 7.4 Gauges for Product Spaces 7.5 Gauges for Infinite‐dimensional Spaces 7.6 Higher‐dimensional Brownian Motion 7.7 Infinite Products of Infinite Products Notes Chapter 8: Quantum Field Theory 8.1 Overview of Feynman Integrals 8.2 Path Integral for Particle Motion 8.3 Action Waves 8.4 Interpretation of Action Waves 8.5 Calculus of Variations 8.6 Integration Issues 8.7 Numerical Estimate of Path Integral 8.8 Free Particle in Three Dimensions 8.9 From Particle to Field 8.10 Simple Harmonic Oscillator 8.11 A Finite Number of Particles 8.12 Continuous Mass Field Notes Chapter 9: Quantum Electrodynamics 9.1 Electromagnetic Field Interaction 9.2 Constructing the Field Interaction Integral 9.3 ‐Complete Integral Over Histories 9.4 Review of Point‐Cell Structure 9.5 Calculating Integral Over Histories 9.6 Integration of a Step Function 9.7 Regular Partition Calculation 9.8 Integrand for Integral over Histories 9.9 Action Wave Amplitudes 9.10 Probability and Wave Functions Notes

9 Part III: Appendices Chapter 10: Appendix 1: Integration 10.1 Monstrous Functions 10.2 A Non‐monstrous Function 10.3 Riemann‐complete Integration 10.4 Convergence Criteria 10.5 “I would not care to fly in that plane” Notes Chapter 11: Appendix 2: Theorem 63 11.1 Fresnel's Integral 11.2 Theorem 188 of [MTRV] 11.3 Some Consequences of Theorem 63 Fallacy Notes Chapter 12: Appendix 3: Option Pricing 12.1 American Options 12.2 Asian Options Notes Chapter 13: Appendix 4: Listings13.1 Theorems 13.2 Examples 13.3 Definitions 13.4 Symbols

10 Bibliography

11 Index

12 End User License Agreement

1 Chapter 2 Table 2.1 Distribution of payouts. Table 2.2 Relative frequency table of distribution of weights. Table 2.3 Calculation of mean and standard deviation. Table 2.4 UD sample paths for processes Table 2.5 Calculations for two UD sample paths for processes

2 Chapter 8Table 8.1 List of four cells forming a regular partition of .Table 8.2 List of representative co‐ordinates for step function calculation.

3 Chapter 9Table 9.1 Elements of calculation of Riemann sum over a regular partition.

4 Chapter 12Table 12.1 Glanbia share prices, 3 August 1991 to 3 September 2011.

1 Chapter 10Figure 10.1 Figure 10.2 Figure 10.3 Figure 10.4

2 Chapter 12Figure 12.1 Figure 12.2 Figure 12.3 , , Figure 12.4 Figure 12.5 Figure 12.6 , , Figure 12.7 moving averages.Figure 12.8 Exponential regressionFigure 12.9 All moving SD'sFigure 12.10 Option values

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2 Table of Contents

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