By 苏剑林 | May 30, 2016
Previously, I promised to share my graduation thesis for everyone's criticism and correction, but I have been lazy about taking action. In fact, the main content of the thesis consists of some introductory-level material on path integrals, titled "Path Integral Methods for Random Walk, Stochastic Differential Equations, and Partial Differential Equations." My abstract was written as follows:
Starting from the random walk model, this paper obtains general results concerning the random walk model. Then, based on the random walk model, path integrals are introduced. Through the path integral method, the transformation between random walks, stochastic differential equations, and parabolic partial differential equations is realized, and some calculation cases are provided.
The path integral method is a form of quantum theory, but in practice, it can be abstracted into a useful mathematical tool. The primary method used in this paper is this abstracted path integral. Furthermore, quantum mechanics features a quite typical parabolic partial differential equation—the Schrödinger equation—which physicists have studied extensively with numerous results. Stochastic differential equations are an extension of differential equations with important applications in physics, engineering, finance, and many other fields, and there are many research methods in this area. Finally, the random walk is a simple and important model that serves as the foundation for many diffusion models and is easy to simulate using computers. Therefore, realizing the transformation between these three is highly meaningful.
This paper contains some new content, such as asymmetric random walks, which are rarely studied in existing literature, and a more detailed introduction to path integrals, which is often vague in current sources. It is intended for the reference of enthusiasts, hoping that this approach will allow readers to understand path integrals in a simpler and clearer way. However, this paper is mainly descriptive, aiming to promote path integral methods within China. Internationally, path integral methods have received considerable attention; they originated in quantum mechanics but their applications are no longer limited to it, as seen in reference [1]. Thus, promoting path integral methods and increasing Chinese-language materials on path integrals is a meaningful and necessary endeavor.
All derivations and examples in this paper are based on the one-dimensional case; corresponding multi-dimensional problems can be calculated similarly.
The general content is as follows in the table of contents:
1 Random Walk
1.1 Model Introduction ................................................ 1
1.2 Asymmetric Random Walk ............................................. 2
1.3 Simplified Form ................................................ 3
1.4 Computer Simulation ............................................... 3
2 Path Integral 4
2.1 From Point Probability to Path Probability ....................................... 4
2.2 Summing over Paths ............................................. 5
2.3 Path Integral of Parabolic Equations .......................................... 5
2.4 From Path Integral to Partial Differential Equation ....................................... 7
2.5 Some Examples ................................................ 7
2.5.1 Most Probable Path ........................................... 7
2.5.2 Quadratic Action .......................................... 8
2.5.3 Perturbation Expansion ............................................ 8
3 Stochastic Differential Equations 9
3.1 Concepts ................................................... 9
3.2 Linear Stochastic Differential Equations ........................................... 9
3.3 Calculating the Jacobian Determinant ........................................... 10
3.4 Path Integral Method .............................................. 12
4 Some Examples 13
4.1 Stock Price Model .............................................. 13
5 Thesis Summary 14
References 15
I do not plan to directly publish the PDF, but will instead post it on this blog with slight modifications. Since the blog's formatting differs somewhat from the original LaTeX, it will take some time to modify. This series is positioned as an introductory tutorial on path integrals, so some parts that were not detailed in the original thesis will be explained further, making it even more detailed than the original. However, since the thesis was required to be complete, some content may overlap with existing articles on the blog; I ask for the readers' understanding.
References:
[1] Hagen Kleinert; Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets 5th Edition [M]; World Book 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; Introduction to Path Integrals and Quantum Physics: An Introduction to Modern 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 Book Publishing Company