This post provides a gentle introduction to stochastic calculus, focusing on the Ito integral. It explains the motivation behind needing a new type of calculus for random processes like Brownian motion, highlighting its non-differentiable nature. The post defines the Ito integral, emphasizing its difference from the Riemann integral due to the non-zero quadratic variation of Brownian motion. It then introduces Ito's Lemma, a crucial tool for manipulating functions of stochastic processes, and illustrates its application with examples like geometric Brownian motion, a common model in finance. Finally, the post briefly touches on stochastic differential equations (SDEs) and their connection to partial differential equations (PDEs) through the Feynman-Kac formula.
The blog post "Kelly Can't Fail" argues against the common misconception that the Kelly criterion is dangerous due to its potential for large drawdowns. It demonstrates that, under specific idealized conditions (including continuous trading and accurate knowledge of the true probability distribution), the Kelly strategy cannot go bankrupt, even when facing adverse short-term outcomes. This "can't fail" property stems from Kelly's logarithmic growth nature, which ensures eventual recovery from any finite loss. While acknowledging that real-world scenarios deviate from these ideal conditions, the post emphasizes the theoretical robustness of Kelly betting as a foundation for understanding and applying leveraged betting strategies. It concludes that the perceived risk of Kelly is often due to misapplication or misunderstanding, rather than an inherent flaw in the criterion itself.
The Hacker News comments discuss the limitations and practical challenges of applying the Kelly criterion. Several commenters point out that the Kelly criterion assumes perfect knowledge of the probability distribution of outcomes, which is rarely the case in real-world scenarios. Others emphasize the difficulty of estimating the "edge" accurately, and how even small errors can lead to substantial drawdowns. The emotional toll of large swings, even if theoretically optimal, is also discussed, with some suggesting fractional Kelly strategies as a more palatable approach. Finally, the computational complexity of Kelly for portfolios of correlated assets is brought up, making its implementation challenging beyond simple examples. A few commenters defend Kelly, arguing that its supposed failures often stem from misapplication or overlooking its long-term nature.
Summary of Comments ( 4 )
https://news.ycombinator.com/item?id=43160779
HN users generally praised the clarity and accessibility of the introduction to stochastic calculus. Several appreciated the focus on intuition and the gentle progression of concepts, making it easier to grasp than other resources. Some pointed out its relevance to fields like finance and machine learning, while others suggested supplementary resources for deeper dives into specific areas like Ito's Lemma. One commenter highlighted the importance of understanding the underlying measure theory, while another offered a perspective on how stochastic calculus can be viewed as a generalization of ordinary calculus. A few mentioned the author's background, suggesting it contributed to the clear explanations. The discussion remained focused on the quality of the introductory post, with no significant dissenting opinions.
The Hacker News post titled "Introduction to Stochastic Calculus" linking to https://jiha-kim.github.io/posts/introduction-to-stochastic-calculus/ has generated several comments discussing various aspects of the topic and the article itself.
Several commenters praise the clarity and accessibility of the introductory article. One user appreciates the author's approach of explaining complex concepts in a simple manner, highlighting the use of clear language and helpful visualizations. They specifically mention the explanation of Brownian motion as being particularly well-done.
Another commenter delves into the practical applications of stochastic calculus, mentioning its use in fields like finance (for option pricing) and physics (for modeling random processes). This commenter expands on the finance application by pointing out how stochastic calculus helps model the unpredictable nature of stock prices.
A further comment chain discusses the challenges inherent in learning stochastic calculus, with one user mentioning the steep prerequisites involving advanced probability theory and calculus. Another user responds by suggesting alternative learning resources and emphasizing the importance of understanding the underlying concepts rather than just memorizing formulas. This thread also touches on the importance of measure theory for a deep understanding of the subject.
One commenter questions the article's statement about integrating over Brownian motion paths, sparking a discussion about the technicalities of defining such integrals and the role of Itô calculus. This thread provides a deeper dive into the mathematical nuances of stochastic integration.
Another commenter notes the article's brevity and expresses hope for the author to expand on certain topics, such as the connection between stochastic differential equations and partial differential equations (specifically the Feynman-Kac formula). This comment highlights the desire for further exploration of advanced topics within the field.
Finally, a few commenters share additional resources, including textbooks and online courses, for those interested in further studying stochastic calculus. These recommendations provide valuable pointers for readers looking to delve deeper into the subject matter.