PAT MULDOWNEY, PHD, was a lecturer at the Magee Business School at Ulster University for more than twenty years. He has published extensively in his areas of expertise: financial mathematics, random variation, Feynman path integrals, and integration theory.

• I Stochastic Calculus 231 Stochastic Integration 252 Random Variation 372.1 What is Random Variation? 372.2 Probability and Riemann Sums 402.3 A Basic Stochastic Integral 422.4 Choosing a Sample Space 502.5 More on Basic Stochastic Integral 523 Integration and Probability 553.1 -Complete Integration 553.2 Burkill-complete Stochastic Integral 623.3 The Henstock Integral 633.4 Riemann Approach to Random Variation 673.5 Riemann Approach to Stochastic Integrals 704 Stochastic Processes 794.1 From Rn to Rª 794.2 Sample Space RT with T Uncountable 874.3 Stochastic Integrals for Example 12 924.4 Example 12 974.5 Review of Integrability Issues 1045 Brownian Motion 1075.1 Introduction to Brownian Motion 1075.2 Brownian Motion Preliminaries 1145.3 Review of Brownian Probability 1175.4 Brownian Stochastic Integration 1205.5 Some Features of Brownian Motion 1275.6 Varieties of Stochastic Integral 1306 Stochastic Sums 1396.1 Review of Random Variability 1406.2 Riemann Sums for Stochastic Integrals 1426.3 Stochastic Sum as Observable 1456.4 Stochastic Sum as Random Variable 1466.5 Introduction to RT(dXs)2 = t 1496.6 Isometry Preliminaries 1516.7 Isometry Property for Stochastic Sums 1536.8 Other Stochastic Sums 1576.9 Introduction to Itô's Formula 1626.10 Itô's Formula for Stochastic Sums 1646.11 Proof of Itô's Formula 1656.12 Stochastic Sums or Stochastic Integrals? 167II Field Theory 1737 Gauges for Product Spaces 1757.1 Introduction 1757.2 Three-dimensional Brownian Motion 1757.3 A Structured Cartesian Product Space 1787.4 Gauges for Product Spaces 1817.5 Gauges for Infinite-dimensional Spaces 1847.6 Higher-dimensional Brownian Motion 1917.7 Infinite Products of Infinite Products 1968 Quantum Field Theory 2038.1 Overview of Feynman Integrals 2068.2 Path Integral for Particle Motion 2108.3 Action Waves 2128.4 Interpretation of Action Waves 2158.5 Calculus of Variations 2178.6 Integration Issues 2218.7 Numerical Estimate of Path Integral 2288.8 Free Particle in Three Dimensions 2368.9 From Particle to Field 2408.10 Simple Harmonic Oscillator 2458.11 A Finite Number of Particles 2518.12 Continuous Mass Field 2579 Quantum Electrodynamics 2659.1 Electromagnetic Field Interaction 2659.2 Constructing the Field Interaction Integral 2709.3 -Complete Integral Over Histories 2739.4 Review of Point-Cell Structure 2789.5 Calculating Integral Over Histories 2799.6 Integration of a Step Function 2839.7 Regular Partition Calculation 2869.8 Integrand for Integral over Histories 2889.9 Action Wave Amplitudes 2919.10 Probability and Wave Functions 295III Appendices 30310 Appendix 1: Integration 30710.1 Monstrous Functions 30810.2 A Non-monstrous Function 30910.3 Riemann-complete Integration 31310.4 Convergence Criteria 31810.5 \I would not care to y in that plane" 32411 Appendix 2: Theorem 63 32511.1 Fresnel's Integral 32511.2 Theorem 188 of [MTRV] 33011.3 Some Consequences of Theorem 63 Fallacy 33512 Appendix 3: Option Pricing 33712.1 American Options 33712.2 Asian Options 34413 Appendix 4: Listings 35713.1 Theorems 35713.2 Examples 35813.3 Definitions 36013.4 Symbols 360

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