Stochastic Calculus Vol I by Steven Shreve
The purpose of getting into this book was having to deal with a lackluster of stochastic probability since I started my Master’s degree. The book came out of my advisor Dr. Yerkin Kitapbayev recommendation. The recommendation was very mcuh in its place. My previous knowledge was accumulated with Statistical Inference by Berger, Intro to Probability by Feller, and Computational Methods in Finance by Ali Hirsa. The issue was that I’ve learned a part of each book without much application to a certain project, just ‘good-to-know’ ordeal. Now, faced with the need to understand the concepts in detail for my PhD, reading a book front to back seemed necessary.
The book starts off simple. Explains probability with spaces, random variables, and distributions with a well written manner. Online videos and texts don’t seem to capture the intuition that Shreve put out clearly in the book. Seems like it was just enough to know the next theorems that he sets later in the book. The binomial model and coin tosses was a nice way to ease into options and stock variations. What really caught my attention that I was not aware of explicitly was the change of variables in the form of state prices. I have to then seek out some background knowledge on measure theory and went over Measure Theory Youtube vidoes.
The subject got more difficult as I kept getting deeper to understand it further. I decided this wasn’t a sustainable path as I just needed some key points. Nonetheless, it seemed like a magic trick when the concept sank in. You just make up a variable, Z, that is the the probability space over risk neutral world divided by the probability space in the actual world. Viola, you can start using this variable to understand actual world probabilities. The random variable also had a distribution to understand and naturally its expected value. Another magical moment occurs right after when we’re introduced to the algorithm to maximize a utility function subject to constraints with a stochastic underlying process. This reminded me of the research that I’m ultimately doing for the PhD, just trying to maximize wealth given a set of assets and utility function. This looks like a Lagrangian multiplier equation and that’s what the author solves. I however have not seen this method for a long time and needed to review the whole of Calc. 3 to understand the concept. After some days, I go back and try to solve problems in chapter 3, and recommend anyone getting into this book to do them. This was my favorite chapter of the book.
The remaining chapters were covered during my studies in probability (random walk chapter) and also my CFA exams (American Options & Interest Rate dependent). Now I’ll move on to Vol II.
