A stand-alone introduction to specific integration problems in the probabilistic theory of stochastic calculus
Picking up where his previous book, A Modern Theory of Random Variation, left off, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics introduces readers to particular problems of integration in the probability-like theory of quantum mechanics.
Written as a motivational explanation of the key points of the underlying mathematical theory, and including ample illustrations of the calculus, this book relies heavily on the mathematical theory set out in the author’s previous work. That said, this work stands alone and does not require a reading of A Modern Theory of Random Variation in order to be understandable.
Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics takes a gradual, relaxed, and discursive approach to the subject in a successful attempt to engage the reader by exploring a narrower range of themes and problems.
Organized around examples with accompanying introductions and explanations, the book covers topics such as:- Stochastic calculus, including discussions of random variation, integration and probability, and stochastic processes- Field theory, including discussions of gauges for product spaces and quantum electrodynamics- Robust and thorough appendices, examples, illustrations, and introductions for each of the concepts discussed within- An introduction to basic gauge integral theory (for those unfamiliar with the author’s previous book)
The methods employed in this book show, for instance, that it is no longer necessary to resort to unreliable “Black Box” theory in financial calculus; that full mathematical rigor can now be combined with clarity and simplicity. Perfect for students and academics with even a passing interest in the application of the gauge integral technique pioneered by R. Henstock and J. Kurzweil, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics is an illuminating and insightful exploration of the complex mathematical topics contained within.
Table of Contents
I Stochastic Calculus 23
1 Stochastic Integration 25
2 Random Variation 37
2.1 What is Random Variation? 37
2.2 Probability and Riemann Sums 40
2.3 A Basic Stochastic Integral 42
2.4 Choosing a Sample Space 50
2.5 More on Basic Stochastic Integral 52
3 Integration and Probability 55
3.1 -Complete Integration 55
3.2 Burkill-complete Stochastic Integral 62
3.3 The Henstock Integral 63
3.4 Riemann Approach to Random Variation 67
3.5 Riemann Approach to Stochastic Integrals 70
4 Stochastic Processes 79
4.1 From Rn to Rª 79
4.2 Sample Space RT with T Uncountable 87
4.3 Stochastic Integrals for Example 12 92
4.4 Example 12 97
4.5 Review of Integrability Issues 104
5 Brownian Motion 107
5.1 Introduction to Brownian Motion 107
5.2 Brownian Motion Preliminaries 114
5.3 Review of Brownian Probability 117
5.4 Brownian Stochastic Integration 120
5.5 Some Features of Brownian Motion 127
5.6 Varieties of Stochastic Integral 130
6 Stochastic Sums 139
6.1 Review of Random Variability 140
6.2 Riemann Sums for Stochastic Integrals 142
6.3 Stochastic Sum as Observable 145
6.4 Stochastic Sum as Random Variable 146
6.5 Introduction to RT(dXs)2 = t 149
6.6 Isometry Preliminaries 151
6.7 Isometry Property for Stochastic Sums 153
6.8 Other Stochastic Sums 157
6.9 Introduction to Itô's Formula 162
6.10 Itô's Formula for Stochastic Sums 164
6.11 Proof of Itô's Formula 165
6.12 Stochastic Sums or Stochastic Integrals? 167
II Field Theory 173
7 Gauges for Product Spaces 175
7.1 Introduction 175
7.2 Three-dimensional Brownian Motion 175
7.3 A Structured Cartesian Product Space 178
7.4 Gauges for Product Spaces 181
7.5 Gauges for Infinite-dimensional Spaces 184
7.6 Higher-dimensional Brownian Motion 191
7.7 Infinite Products of Infinite Products 196
8 Quantum Field Theory 203
8.1 Overview of Feynman Integrals 206
8.2 Path Integral for Particle Motion 210
8.3 Action Waves 212
8.4 Interpretation of Action Waves 215
8.5 Calculus of Variations 217
8.6 Integration Issues 221
8.7 Numerical Estimate of Path Integral 228
8.8 Free Particle in Three Dimensions 236
8.9 From Particle to Field 240
8.10 Simple Harmonic Oscillator 245
8.11 A Finite Number of Particles 251
8.12 Continuous Mass Field 257
9 Quantum Electrodynamics 265
9.1 Electromagnetic Field Interaction 265
9.2 Constructing the Field Interaction Integral 270
9.3 -Complete Integral Over Histories 273
9.4 Review of Point-Cell Structure 278
9.5 Calculating Integral Over Histories 279
9.6 Integration of a Step Function 283
9.7 Regular Partition Calculation 286
9.8 Integrand for Integral over Histories 288
9.9 Action Wave Amplitudes 291
9.10 Probability and Wave Functions 295
III Appendices 303
10 Appendix 1: Integration 307
10.1 Monstrous Functions 308
10.2 A Non-monstrous Function 309
10.3 Riemann-complete Integration 313
10.4 Convergence Criteria 318
10.5 \I would not care to y in that plane" 324
11 Appendix 2: Theorem 63 325
11.1 Fresnel's Integral 325
11.2 Theorem 188 of [MTRV] 330
11.3 Some Consequences of Theorem 63 Fallacy 335
12 Appendix 3: Option Pricing 337
12.1 American Options 337
12.2 Asian Options 344
13 Appendix 4: Listings 357
13.1 Theorems 357
13.2 Examples 358
13.3 Definitions 360
13.4 Symbols 360
