I previously promised to share my graduation thesis for everyone to critique and provide feedback, but I have been procrastinating. In fact, the main content of the graduation thesis consists of some introductory-level material on path integrals, titled "Path Integral Methods for Random Walks, Stochastic Differential Equations, and Partial Differential Equations". My abstract was written as follows:
This paper starts from the random walk model and obtains general results regarding it; then, based on the random walk model, path integrals are introduced. Through the path integral method, the mutual 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 formulation of quantum theory, but in fact, it can be abstracted as a useful mathematical tool. The main method of this paper is precisely this abstracted path integral. Secondly, there is a quite typical parabolic partial differential equation in quantum mechanics—the Schrödinger equation. Physicists have conducted extensive research on it and achieved 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 yet important model; it is the foundation of many diffusion models and possesses characteristics that make it easy to simulate using computers. Therefore, realizing the transformation between these three is of great significance.
This paper contains some new content, such as the study of asymmetric random walks, which is relatively rare in existing literature, and an introduction to path integrals that is clearer than the often ambiguous descriptions found in current texts. It is intended for reference by enthusiasts, hoping that through this approach, readers can understand path integrals in a more concise and clear manner. However, this paper is primarily descriptive, aiming to promote the path integral method domestically. Abroad, the path integral method has received considerable attention; it originated from quantum mechanics, but its applications are no longer limited to quantum mechanics, as seen in reference [1]. Therefore, promoting the path integral method and increasing Chinese-language materials on path integrals is a very meaningful and necessary task.
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 shown 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 Integrals 4
2.1 From Point Probability to Path Probability ....................................... 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 Calculation 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 ........................................... 10
3.4 Path Integral Method .............................................. 12
4 Some Examples 13
4.1 Stock Price Model .............................................. 13
5 Thesis Review 14
References 15
I do not intend to publish the PDF directly; instead, I will publish it on the blog with slight modifications. Since the layout of the blog differs from the original LaTeX, it will take some time to adjust. This series is positioned as an introductory tutorial on path integrals. Therefore, I will provide more explanations for some parts that were not detailed in the original thesis, making it even more comprehensive than the original. However, since the thesis required a certain level of completeness, some content may overlap with existing articles on the blog; I hope readers will understand.
References
] Hagen Kleinert; Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th Edition [M]; World Book Publishing Company
] Papoulis A., Pillai S.U.; Probability, Random Variables and Stochastic Processes (4th Edition) [M]; Xi’an Jiaotong University Press
] Gregory F. Lawler; Random Walk and the Heat Equation [J]
] Sheldon M. Ross (Author), Gong Guanglu (Translator); Stochastic Processes [M]; Mechanical Industry Press
] Feynman; Quantum Mechanics and Path Integrals [M]; Higher Education Press
] Hou Boyuan, Yun Guohong, Yang Zhanying; Introduction to Path Integrals and Quantum Physics: Introduction to Modern Advanced Quantum Mechanics [M]; Science Press
] M Chaichian, A Demichev; Path Integrals in Physics: Volume I Stochastic Processes and Quantum Mechanics
] Carson C. Chow, Michael A. Buice; Path Integral Methods for Stochastic Differential Equations [J]
] Horacio S. Wio; Application of Path Integration to Stochastic Processes: An Introduction [J]
] Belal E. Baaquie; Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates [M]; World Book Publishing Company
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