Path Integral Series: 1. My Graduation Thesis

路径积分系列：1.我的毕业论文 苏剑林 · 2016-05-30 · Mathematics path integral

I've long promised to share my graduation thesis so that everyone could critique and correct it, but kept procrastinating. In fact, the main content of the thesis is a bit of introductory-level material on path integrals, titled "Path Integral Methods for Random Walks, Stochastic Differential Equations, and Partial Differential Equations." My abstract reads as follows:

Starting from the random walk model, this paper derives some general results about random walk models; it then introduces the path integral based on the random walk model, and uses the path integral method to establish mutual transformations among random walks, stochastic differential equations, and parabolic differential equations, along with some worked examples.
The path integral method is a formulation of quantum theory, but in fact it can be abstracted into a useful mathematical tool, and it is precisely this abstracted path integral that forms the main method of this paper. Second, quantum mechanics contains a rather typical parabolic partial differential equation—the Schrödinger equation—which physicists have studied extensively, yielding a wealth of results. Stochastic differential equations, meanwhile, are an extension of ordinary differential equations with important applications in physics, engineering, finance, and many other fields, and there are likewise many research methods in this area. Finally, the random walk is a simple yet important model that underlies many diffusion models and has the property of being easy to simulate on a computer. For these reasons, establishing the interconversion among these three is a meaningful undertaking.
This paper contains some new material, such as a discussion of asymmetric random walks—a topic relatively underexplored in the existing literature—as well as a clearer introduction to path integrals than is commonly found elsewhere, which may be of use to fellow enthusiasts. I hope that through this presentation, some readers will be able to understand path integrals in a more concise and clear way. However, this paper is mainly expository in nature, aiming to promote the path integral method domestically. Abroad, the path integral method has received considerable attention; it originates from quantum mechanics, but its applications are no longer confined to quantum mechanics, as seen for instance in reference [1]. Therefore, promoting the path integral method and increasing the availability of Chinese-language materials on path integrals is a meaningful and necessary endeavor.
All derivations and examples in this paper are presented in one dimension; the corresponding multidimensional problems can be computed analogously.

The general contents are as follows (table of contents):

1 Random Walks
1.1 Overview of the Model ................................................ 1
1.2 Asymmetric Random Walks............................................. 2
1.3 Simplified Form ................................................ 3
1.4 Computer Simulation............................................... 3
2 Path Integrals 4
2.1 From the Probability of a Point to the Probability of a Path ....................................... 4
2.2 Summing Over Paths............................................. 5
2.3 Path Integrals for Parabolic Equations.......................................... 5
2.4 From Path Integrals to Partial Differential Equations ....................................... 7
2.5 Some Worked Examples ................................................ 7
2.5.1 The Most Probable Path........................................... 7
2.5.2 Quadratic Action.......................................... 8
2.5.3 Perturbative Expansion ............................................ 8
3 Stochastic Differential Equations 9
3.1 Concepts................................................... 9
3.2 Linear Stochastic Differential Equations ........................................... 9
3.3 Computing the Jacobian Determinant ........................................... 10
3.4 Path Integral Method.............................................. 12
4 Some Examples 13
4.1 A Stock Price Model.............................................. 13
5 Literature Review 14
References 15

I don't plan to publish the PDF directly, but rather to publish it on the blog with some modifications. Since the blog's formatting differs somewhat from the original LaTeX, this will take a certain amount of time to adapt. This series is intended as an introductory tutorial on path integrals, so I'll elaborate further on parts of the original thesis that were left somewhat underexplained, meaning it will actually end up more detailed than the original thesis. However, since the thesis was written to meet requirements of completeness, some of the content may overlap with articles already on the blog—I ask readers' understanding on this point.

References:

[1] Hagen Kleinert; Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th ed. [M]; World Scientific Publishing Company
[2] Papoulis A., Pillai S.U.; Probability, Random Variables, and Stochastic Processes (4th ed.) [M]; Xi'an Jiaotong University Press
[3] Gregory F. Lawler; Random Walk and the Heat Equation [J]
[4] Sheldon M. Ross (author), Gong Guanglu (translator); Stochastic Processes [M]; China Machine Press
[5] Feynman; Quantum Mechanics and Path Integrals [M]; Higher Education Press
[6] Hou Boyuan, Yun Guohong, Yang Zhanying; An Introduction to Path Integrals and Quantum Physics: A Modern Primer on Advanced Quantum Mechanics [M]; Science Press
[7] M Chaichian, A Demichev; Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics
[8] Carson C. Chow, Michael A. Buice; Path Integral Methods for Stochastic Differential Equations [J]
[9] Horacio S. Wio; Application of Path Integration to Stochastic Processes: An Introduction [J]
[10] Belal E. Baaquie; Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates [M]; World Scientific Publishing Company