Overview of basic results in Stochastic Calculus

Question

Are there some good overviews of basic facts about Stochastic Integrals and Stochastic Calculus? These can be in the form of resources (preferably accessible online) as well as directly writing out these results as answers.

If possible, it would be helpful to link to proofs of the results. These can be external proofs or proofs on the site.

This question was inspired by other similar [big-list] questions including overviews on basic results on images and preimages and on basic results in cardinal arithmetic. See the answers at these links for inspiration on the types of responses that would be suitable for this question.

Edit: I have now offered a bounty to raise awareness of the question and encourage strong answers for future reference.

Edit 2: I have posted an answer with links to lecture notes and blogs. However, I still welcome other answers. I am especially looking for an answer with a direct list of results. Although the answer I have posted with links is a good start, it would be better to have the properties, theorems, etc. directly written out here. I won't be accepting an answer until at least the day after the bounty expires.

Answer 1

I'll include an answer with books. If you're looking for a comprehensive overview of basic results, then these books will for sure do the job. They do way more actually.

1. Medvegyev, P. (2007). Stochastic integration theory (Vol. 14). OUP Oxford.

Goes through the general theory of stochastic integration; It provides a clear exposition of essential concepts such as local martingales, semimartingales and quadratic variation; It also gives very good intuition on some measure theoretic considerations; Most classical theorems of Stochastic Calculus can be found here in great generality; Overall, one of the best books in this list, at least I think so.

2. Schilling, R. L., & Partzsch, L. (2014). Brownian motion: an introduction to stochastic processes. Walter de Gruyter GmbH & Co KG.

Covers Brownian Motion in a complete and comprehensive way; If the goal is to deeply understand Brownian motion and obtain a solid, yet introductory, understanding of stochastic integration, then this is a great book; It goes through the $L^2$-theory of stochastic integrals, i.e. the integrators are $L^2$-martingales.

3. Chung, K. L., & Williams, R. J. (1990). Introduction to stochastic integration (Vol. 2). Boston: Birkhäuser.

Offers an easy to read account of the stochastic integral; The integrators considered are at most continuous local martingales, so it's not as general as 1. for instance; The Preliminaries chapter gives a nice idea of the pre-requisites to study stochastic calculus;

4. Cohen, S. N., & Elliott, R. J. (2015). Stochastic calculus and applications (Vol. 2). New York: Birkhäuser.

This is an interesting book if one is interested in other topics that are directly linked to stochastic calculus: jump processes, backward stochastic differential equations, optimal control, and so on. The first part of the book provides a comprehensive overview of the measure theoretic pre-requisites;

5. Klebaner, F. C. (2012). Introduction to stochastic calculus with applications. World Scientific Publishing Company.

This books provides a rather concise overview of stochastic calculus. If the goal is to have a global idea of applications (to finance, engineering, biology,...), then this book does the job. Includes some nice motivations;

6. Karatzas, I., & Shreve, S. (2012). Brownian motion and stochastic calculus (Vol. 113). Springer Science & Business Media.

A classic. That said, I would only recommend this book to someone with decent mathematical foundations in probability, analysis and measure theory. It is a book for someone with proper mathematical training. I recommend having a go at this book after reading some other introductory text on stochastic calculus.

7. Le Gall, J. F. (2016). Brownian motion, martingales, and stochastic calculus. springer publication.

It offers a very rigorous presentation of stochastic integration; I highly recommend this book to trained mathematicians; Advanced undergraduates might benefit from the book as well; I like this book because it's not too "exhaustive"; The writing is precise and straight to the point.

8. Introduction to Stochastic Integration, Hui-Hsiung Kuo

My favourite introductory book. The most pleasant read in this list.

9. Arguin, L. P. (2021). A first course in stochastic calculus (Vol. 53). American Mathematical Society.

Great introductory book as well; Recommend it for undergraduate students; It's full of intuition; It also includes very nice numerical projects; Appropriate for those who want to know stochastic calculus to use it solely as a tool. I particularly enjoy the sections dedicated to change of measure.

10. Stochastic Integration and Differential Equations, Protter, Philip

Another classic, and not the easiest read I'd say. It has a careful treatment of semimartingales. Similarly to 6. I'd recommend using this book after reading some other introductory text.

11. Oksendal, B. (2013). Stochastic differential equations: an introduction with applications. Springer Science & Business Media.

Classic number 3. Compared to the other "classics" I'd say this one is more readable, while maintaining a decent level of rigour. Still I'd say this is a graduate level book.

12. Evans, L. C. (2012). An introduction to stochastic differential equations (Vol. 82). American Mathematical Soc..

Covers most elementary facts in stochastic calculus; Goes through several of its uses; Nice complementary book overall.

To sum up: for introductions, I highly recommend 8. For more advanced textbooks, I recommend 6, 10 and 7. Personally, I really enjoyed studying from 1 and 2.

Answer 2

I will start off with an answer dedicated to publically available online lecture notes and blogs. I still welcome other answers - especially those that contain explicit results as opposed to linked resources. I am making this answer Community Wiki so that others can contribute and allow this list to expand into a comprehensive resource.

Lecture Notes

• Stochastic Calculus (University of Cambridge)

• An Introductory Course on Stochastic Calculus (University of Melbourne)

• Stochastic Calculus for Finance (Carnegie Mellon University)

• Stochastic Calculus: An introduction with applications (University of Chicago)

Blogs

• Almost Sure (George Lowther)
• Stochastic Calculus: Navigating Risk Neutral Measures (FasterCapital)