布朗运动与随机演算
Brownian motion and stochastic calculus
Ioannis Karatzas、Steven E. Shreve · 初版 1988 年
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本书专为研究生阶段的 stochastic processes 课程编写。适合熟悉 measure-theoretic probability 和 discrete-time processes 理论、并准备进一步探索 continuous-time stochastic processes 的读者阅读。本书选用的主要阐述载体是 Brownian motion，它被呈现为连续时间下 Markov process 与 martingale 的经典范例。作者展示了如何借助 stochastic integration 与 random time change，将所有 continuous martingales 以及诸多 continuous Markov processes 表达为 Br
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……- Lamberton 和 Lapeyre 的《金融随机分析导论》（一本很棒的小入门书） - Tomas Bjork 的《连续时间套利理论》（展开节奏较慢，覆盖面更广） - Karatzas 和 Shreve 的《布朗运动与随机分析》以及《数理金融方法》（理论基础扎实，适合数学底子更好的人）。之前有一本不用测度论讨论金融模型的好书……
…- "Introduction to Stochastic Calculus Applied to Finance" by Lamberton and Lapeyre (Nice little intro) - "Arbitrage Theory in Continuous Time" by Tomas Bjork (Slower discussion, larger breadth) - "Brownian Motion and Stochastic Calculus" and "Methods of Mathematical Finance" by Karatzas and Shreve (Solid theoretical foundation, for the more mathematically inclined) There was a good non-measure theoretic discussion of financial models…
……Breiman 的《概率论》比 Neveu 的好读。钟开莱那本竞品书也是。还有其他一些。有一些更进阶的教材，比如 Ioannis Karatzas 和 Steven E. Shreve 的《布朗运动与随机分析（第二版）》。祝好运！
…Breiman's Probability is easier to read than Neveu. So, is K. L. Chung's competitive book. And there are others. There is some more advanced material, e.g., Ioannis Karatzas and Steven E.\ Shreve, {\it Brownian Motion and Stochastic Calculus, Second Edition,\/} Good luck!
……泊松过程以及有限马尔可夫链中的主要收敛定理（也是一个线性代数结论）。想了解更多，可以再看看 Breiman 和 Loeve 的相关章节，然后是：Karatzas 和 Shreve 的《布朗运动与随机分析》。
…Poisson process and the main convergence theorem in finite Markov chains (also a linear algebra result). Then for more, note again the relevant sections of Breiman and Loeve and then: Karatzas and Shreve, Brownian Motion and Stochastic Calculus.
……等等。现在你就能看懂电子工程（EE）中对二阶平稳随机过程、数字滤波、功率谱估计等的讨论了。随机过程方面，比如 Karatzas 和 Shreve 的《布朗运动与随机分析》。现在可以开始切入数理金融了。但在数值方法、线性规划、拉格朗日乘数法（一种出人意料的通用技术）、整数规划这些方向上也有很多拓展路线……
…etc. Now will be able to understand EE treatments of second order stationary stochastic processes, digital filtering, power spectral estimation, etc. Stochastic processes, e.g., Karatzas and Shreve. Brownian Motion and Stochastic Calculus. Now can get started on mathematical finance. But there are many side trips available in numerical methods, linear programming, Lagrange multipliers (a surprisingly general technique), integer programming (a…
……Loeve 的学生。可以用这项工作把布朗运动彻底搞明白，这方面的书多得很。在随机过程上还可以更进一步，比如 Ioannis Karatzas 和 Steven E. Shreve 的《布朗运动与随机分析》；R. S. Lipster 和 A. N. Shiryayev 的《随机过程统计》；Ronald K. Getoor 的《马尔可夫过程与位势论》。这些材料都有几十年历史了。有了这些底子，完全可以讲清楚……
…Loeve students. Can use this work to make good sense out of Brownian motion, and there are plenty of books there. And can do more on stochastic processes, e.g., Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus. R. S. Lipster and A. N. Shiryayev, Statistics of Random Processes. Ronald K. Getoor, Markov Processes and Potential Theory. This material is now decades old. With this material it is possible to give…
……需要具备带西格玛代数的测度论、马尔可夫过程、布朗运动、拉东-尼科迪姆定理、条件期望、随机积分等前置知识，例如 Ioannis Karatzas、Steven E. Shreve 的《布朗运动与随机分析》，Robert M. Blumenthal 和 Ronald K. Getoor 的《马尔可夫过程与位势论》。可能还会需要抛物型偏微分方程和随机动态规划。以我的经验，如果具备了……
…prerequisites of measure theory with sigma algebras, Markov processes, Brownian motion, the Radon-Nikodym theorem, conditional expectation, stochastic integration, e.g., Ioannis Karatzas, Steven E. Shreve, Brownian Motion and Stochastic Calculus, Robert M. Blumenthal and Ronald K. Getoor, Markov Processes and Potential Theory. Can also want parabolic partial differential equations and stochastic dynamic programming. In my experience, with a…