The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Economics, Engineering, and Finance
The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Economics, Engineering, and Finance
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DOI：
10.1198/jasa.2002.s242
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发表时间：
2002-12
影响因子：
3.7
通讯作者：
A. McNeil
中科院分区：
文献类型：
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作者：
A. McNeil
通过简单地使用黎曼求和和泰勒公式，以快速的方式。我用这个演示文稿在教学研究生与弱概率背景，但良好的物理直觉和强大的微积分能力，并发现它相当有效，完全令人满意的应用。第3章将进一步探讨路径朗之万解释，其中对奥恩斯坦-乌伦贝克过程的物理解释使我们对布朗运动背后的经典力学有了令人钦佩的理解。我相信，在第3章中描述的布朗运动和扩散在物理学中的这一点和其他有趣的应用，在涵盖随机微分方程的标准教科书中找不到。因此，这本书是一个真正的创新。一个强大的物理后盾，以介绍的主题是普遍只在一部分的书。第4章讨论了当空间步的平方或时间步的平方均为非线性平均时，作为超扩散或次扩散连续时间随机游动的稳定极限而出现的随机过程。这一材料是在一个更数学形式的方式比其他章节。尽管如此，如果这本书是由一个纯粹的概率论者写的，那么它的处理方式并不像人们所期望的那样枯燥。尽管有许多技术计算，作者设法使这个困难的话题成为一个愉快的阅读。第5章是唯一一章处理金融应用程序，为任何没有金融背景的人提供了宝贵的介绍。它包括对主要金融流行语的简短但出色的介绍，这些流行语是理解数学金融论证的基础，沿着对布莱克-斯科尔斯理论的基本处理。与本书的其余部分一致，作者努力使治疗尽可能生动。然而，由于几个原因，这一章没有达到我的期望。首先，我很惊讶地发现，在第2章和第3章中，作者们花了大量的精力来分析扩散过程，却选择了偏微分方程（PDE）方法作为推导布莱克-斯科尔斯方程的主要工具。他们选择提出概率方法和风险中性估值公式，只是作为一种评论。相反，他们可以使用对冲思想和伊藤公式更明确地推导布莱克-斯科尔斯偏微分方程，通过费曼-卡茨公式求解，然后用它以纯粹概率的方式推导布莱克-斯科尔斯公式。在随机过程的背景下，不应该介绍的数学一致性应力随机微积分偏微分方程？例如，Bjork（1998）第6章和第7章对这一主题的处理在这里会更合适。为了进一步理解风险中性估值的力量（作者的介绍并不完全具有启发性），本书将贝内于离散时间美式期权的基本处理，相当于兰伯顿和拉佩尔（1996）优秀的第2章的总结。我之前提到的另一个失望是，物理学家在本章中对数学一致性的处理并没有像我希望看到的那样给我带来启发。这可能是因为我对本章中涉及作者所谓的“经济物理学”最新进展的部分缺乏很好的理解。一个似乎来自经济物理学的模型是基于本书前面提到的截断的列维·莱特模型;它旨在提供对股票价格波动是随机的这一臭名昭著的观察结果的理解。我未能理解这一模型背后的物理原理，部分原因可以归咎于作者不够理想的陈述。本章的其余部分包含了一些关于如何获得所谓的波动率微笑的有趣信息，以及当前对金融危机建模的一些努力，但仍然缺乏本书其余部分的清晰度。尽管我对这本书的金融部分缺乏热情，但我非常喜欢它的其余部分。一个特别有用的特点，特别是考虑到作者打算写一本教科书，是在每一章的结尾都有一个深思熟虑的总结。按字母顺序排列的主题索引是另一个很好的功能。更难把这本书当作教科书的是缺乏练习。因此，这本书可能只适合研究生班，可能只在理科学生的掌握范围内。这将为学生研讨班打下良好的基础，在研讨班中，每个学生的任务是介绍书中的一章，也许还可以查阅每章末尾慷慨提供的一些参考资料。最后，前四章的高质量使这本书非常值得阅读的物理学生寻求介绍连续时间随机过程，以及任何人与扩散过程的良好知识谁可能有兴趣的物理原因证明理论及其扩展到稳定的过程。
by simply using Riemann sums and Taylor’s formula, in an expeditious manner. I have used this presentation in teaching graduate students with a weak probabilistic background but a good physical intuition and strong calculus abilities and have found it quite effective and entirely satisfactory for applications. The pathwise Langevin interpretation is exploited further in Chapter 3, in which the physical interpretation of the Ornstein-Uhlenbeck process gives an admirable understanding of the classical mechanics underlying Brownian motion. I believe that this and other interesting applications of Brownian motion and diffusions to physics described in Chapter 3 are not found in standard textbooks covering stochastic differential equations. As such, this book presents a true innovation. A strong physical backing to the presentation of the topics is prevalent only in a portion of the book. Chapter 4 deals with stochastic processes that arise as stable limits of superdiffusive or subdiffusive continuous-time random walks, when either the squared space step or the time step have in nite mean. This material is presented in a more mathematically formal way than in the other chapters. Nevertheless, the treatment is not as dry as one would expect had the book been written by a pure probabilist. Despite the numerous technical calculations, the authors manage to make this dif cult topic an enjoyable read. Chapter 5, the only chapter dealing with nancial applications, provides an invaluable introduction for any person with no nancial background. It includes a short but excellent presentation of the main nancial buzz words that are the basis for understanding a mathematical nancial argumentation, along with a basic treatment of the Black–Scholes theory. Consistent with the remainder of the book, the authors have strived to make the treatment as lively as possible. However, this chapter did not live up to my expectations for several reasons. First, I was surprised to see that, with all of the effort spent in Chapters 2 and 3 on analyzing diffusion processes, the authors chose to present a partial-differential equations (PDE) method as the main tool for deriving the Black–Scholes equation. They chose to present the probabilistic method, and the risk-neutral valuation formula, only as a kind of remark. Instead, they could have used hedging ideas and Itō’s formula to derive the Black–Scholes PDE more explicitly, solving it via a Feynman–Kac formula and then using that to derive the Black–Scholes formula in a purely probabilistic way. In the context of stochastic processes, should not a presentation of mathematical nance stress stochastic calculus over PDEs? The treatment of this topic in Chapters 6 and 7 of Bjork (1998), for instance, would have been more appropriate here. For a further understanding of the power of risk-neutral valuation (whose presentation by the authors is not entirely enlightening) the book would have bene tted from a basic treatment of American options in discrete time, corresponding to a summary of the excellent Chapter 2 of Lamberton and Lapeyre (1996). My other disappointment, which I alluded to earlier, is that I was not enlightened by the physicists’ treatment of mathematical nance in this chapter in the way that I was hoping to see. This may be due to my lack of a good understanding of the portion of the chapter that deals with recent progress in what the authors call “econophysics.” One model that seems to come from econophysics is based on the truncated Lévy ight, presented earlier in the book; it aims to provide an understanding of the notorious observation that stock price volatility is stochastic. My failure to understand the physics underlying this model can be partially blamed on the authors’ lessthan-optimal presentation. The remainder of the chapter contains some interesting information on how to obtain the so-called volatility smile and on some current efforts to model nancial crashes, but it still lacks the clarity of the remainder of the book. Despite my lack of enthusiasm for the nancial part of this book, I thoroughly enjoyed the remainder of it. One particularly useful feature, especially given the authors’ intent to write a textbook, is the presence of a well-thoughtout summary at the end of each chapter. The alphabetical subject index is another good feature. What makes it more dif cult to consider this book as a textbook is the lack of exercises. As such, this book could be appropriate only for a graduate class, and may only be within the grasp of science students. It would make a good basis for a student seminar class in which each student’s task would be to present a chapter of the book, and perhaps look up some of the references generously provided at the end of each chapter. In the end, the high quality of the rst four chapters make the book well worth reading for physics students seeking an introduction to continuous-time stochastic processes, as well as anyone with a good knowledge of diffusion processes who might be interested in the physical reasons justifying the theory and its extensions to stable processes.