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.
- 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.
- 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.
- 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;
- 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;
- 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;
- 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.
- 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.
- Introduction to Stochastic Integration, Hui-Hsiung Kuo
My favourite introductory book. The most pleasant read in this list.
- 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.
- 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.
- 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.
- 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.
