# Where to begin in approaching Stochastic Calculus?

I have experience in Abstract algebra (up to Galois theory), Real Analysis(baby Rudin except for the measure integral) and probability theory up to Brownian motion(non-rigorous treatment). Is there a suggested direction I can take in order to begin studying stochastic calculus and stochastic differential equations?

• When you say probability, is it the measure theory approach or the classical one? Commented Nov 6, 2012 at 22:20
• most likely measure theory approach, considering this is stochastic calculus I imagine it would have to use measure theory. Commented Nov 6, 2012 at 22:23
• What I meant was, what kind of probability have you done Commented Nov 6, 2012 at 22:23
• classical, It was non-rigorous. I have some idea of the measure theory approach, having worked with (again non-rigorous) martingales in a math finance course. You have a probability space and essentially probability is just a function that maps an event (or set of events) to a real number. Commented Nov 6, 2012 at 22:28

## 5 Answers

I Suggest

For Measure Theory

1. Real Analysis -Royden
2. Measure Theory- Halmos.

For probability theory, Brownian motion and stochastic Calculus

1. "Probability with Martingales" by David Williams.
2. "An Introduction to Probability Theory and Its Applications 1-2" William Feller.
3. "Diffusions, Markov Processes and Martingales:1-2" by Chris Rogers and David Williams.
4. "Introduction to Stochastic Integration" by K. L. Chung, R.J. Williams
5. "Stochastic Differential Equations: An Introduction with Applications" by Bernt Øksendal.

You may also need to learn some Complex Analysis. Although Complex analysis is not essential to learn probability theory and stochastic processes. However, contour integration and Fourier transforms are indispensable tools and it is also one of the most beautiful and useful areas of mathematics.

• do you have some complex analysis book recommendations suited for this purpose? I mean a complex analysis book most helpful for someone studying graduate level probability/statistics. I have undergraduate training in real analysis (baby Rudin) and linear algebra (Valenza). Commented Feb 26, 2020 at 16:50
• @iceberg perhaps take a look at Complex Analysis visualized. Commented Jul 19, 2020 at 19:59

In my experience it is best to get to your goal as quickly as possible to maintain momentum. You could read Royden and then Billingsley and finally start on stochastic calculus. But personally, I would probably run out of steam before long.

I would recommend you read

Jeff Rosenthal's book A First Look at Rigorous Probability.

It is 200 pages long. It is very clearly explained (baby Rudin is all you need). It develops all the measure theory you need in a probability context. It has a lot of easy exercises that build confidence that you understand basic concepts. Half of these have solutions!

The last chapter whets your appetite for stochastic calculus and he gives suggested reading.

• Rosenthal's book actually reads very well. I like it. Thanks for the suggestion. (amazon preview) Commented Nov 7, 2012 at 17:48

As previous answers have indicated, a solid measure-theoretic approach to probability is essential. A book I would strongly recommend for the measure-theoretic approach to probability is:

R. M. Dudley - Real Analysis and Probability

I found it very clear, well-organised and invaluable when I was working on such things.

There are several textbooks out there for finance professionals that do not require heavy understanding of measure theory (don't require any). I think there is an unconscious bias of math professionals to want to learn the theory. But it's totally not required. Just as an engineer can learn to solve integers and solve them well with lots of tricks, never having done theoretical calculus, so can you very easily learn the major methods of stochastic calculus, with no more prereqs than what you already have.

BTW, nothing stops you from going back later and learning a bunch of rigor theory stuff. But why not go now and get the basic manipulational skills. They will do you more good in any sort of finance job. And it preserves option value (decide to invest in the rigor later when you know you need it.)

Actually I would even go further and say that you are more likely to succeed in learning the material, to learn it deeply, and even time efficiently, if you do it the "easy way" first and move only to rigor as a progression.

• How do you "solve integers" ? Commented Mar 14, 2017 at 12:23
• i am guessing that was typo for integrals. Commented Dec 24, 2017 at 21:22
• A useful comment. Commented Nov 7, 2019 at 21:49

I would suggest to first take a course and/or get a book on probability theory, in view of measure theory. It helps to have studied measure theory first but is not necessary.

Some textbook ideas, but in no way an exhaustive list:

• Durrett: Probability: Theory and Examples. I've followed a class using this book, you can find an earlier edition online I think
• Billingsley: Probability and Measure
• Gut: Probability: A graduate course

I have not worked with the last 3 but only consulted for references. Perhaps our more experienced user could have a better say in the matter.

Once you have done that, you can take a class on stochastic calculus in general. That should explore the construction of Brownian motion, the Ito integral, some Stochastic Differential equations and a continuation of martingales that you will have started in course 1. Some books are

• Shreve, and also Steele have books with some financial emphasis
• Karatzas and Shreve's Brownian Motion and Stochastic Calculus has been around a while but might be harsh for a first class

You can then take more advanced class on specific topic such as Stochastic Differential equations. One book that comes to mind is Oksendal's