Books recommendations on stochastic analysis

Question

I'm currently self-study stochastic analysis by reading Rogers' Diffusion,Markov processes and Martingales and other books as supplement.I found that every book had it's own preference and didn't contained all basic theorem about the topic.So I wonder if there is a 'Bible' in stochastic analysis which contains most of the theorem (like Folland in real analysis,Lang in algebra),so that I can use it as a dictionary.By the way,I'd also like to know some books about stochastic analysis with good exercise.

Any help is appreciated.

Answer 1

i think unfortunately there is no such reference like a bible for stochastic calculus. In my experience this topic makes use of several theories, and each of them has entire books on the topics. But some books bring a variety of topics that introduce important concepts that can be studied separately in specialized references.

I'm doing a master degree in mathematical methods in finance and i'll put here some references that i have used along the course to study stochastic calculus. My course have a pratical approach so, if you want somenthing more related with pure mathematics you should maybe go deeper in mesure and integration theory and analysis.

For practical approaches:

1. Stochastic Calculus for Finance I: Binomial asset pricing model and Stochastic Calculus for Finance II: tochastic Calculus for Finance II: Continuous-Time Models. These two books are very good if you want to apply the theory to price derivatives.

2. Stochastic Differential Equations: An Introduction with Applications Bernt Oksanda. I think that this book have a havier math than Sherevs book, and i think its a very good reference. And this book have a lot of solved excercises and this helps a lot if you get stucked in some problem.

3. Brownian Motion Calculus - Ubbo F Wiersema. This one have a very pratical approach and a lot a problems to solve. It have the proofs of the theorems and a lot of pratical applications and solved examples.

4. And finally there is nice free courses like Advanced sthocastic process from MIT open course (https://ocw.mit.edu/courses/15-070j-advanced-stochastic-processes-fall-2013/pages/syllabus/) and Introduction To Stochastic Processes ( https://ocw.mit.edu/courses/18-445-introduction-to-stochastic-processes-spring-2015/). This 2 curses have video classes that are very good.

5. There are some youtube chanels that makes a very good job explaining some conceps of stochatic calculus. I like quantpie a lot(https://www.youtube.com/@quantpie)

For a pure mathematical apporach:

1. . FOLLAND, GB. Real analysis: modern techniques and their applications. 2nd ed. New York, Wiley, 1999
2. ROYDEN, H.L.. Real Analysis. 3rd ed. New York: Macmillan Publishing Company, 1988.
3. BARTLE, R.G.. The elements of integration and Lebesgue measure. New York: John Wiley, 1995

I emphasize that these books are references that i have at my level of knowledge. Which is much lower than that of an experienced specialist, such as a PhD in the matter. So, if you want to go deeper into the subjects, there are certainly other books that I haven't explored yet that go deeper into the subjects, and especially important papers that must be read to understand the foundations of some theories.

I hope it helps you!