I have already taken a couse in Stochastic Calculus. Due to time constraints on many ocassions we had to skip some formalities among the proofs. I'm trying now to fill the gaps left, and I have been searching for a book to do so. My problem is that I haven't found many good references.

I'm intersted in a book (or books) with rigorous treatment of:

  • Brownian Motion (Wiener Process, Wiener Measure and construction)
  • Martingale Theory (Discrete and Continuous, but specially the transition from Discrete to Continuous Time)
  • Stochastic Calculus (Ito Integration)
  • SDE

I have already explored some books such as Karatsas but have found them very dry and almost encyclopedia like, which is something I don't like from books.

Any references (online notes or books) are appreciated. I'm kind of trying to overcome the thought that this subject (Stochastic Calculus) is filled with dry formalities. I'm trying to find a treatment which balances intuition and formality but without feeling dry and devoid of motivation.

By the way I have a good base on measure theory so no problem with it as a prequisite.

Thanks in advance

  • $\begingroup$ Karatzas and Shreve's Stochastic Calculus and Brownian Motion is probably the best written reference on the topic (esp. next to the Russian books like Shiryaev's). Books like Oskendal's SDE are nice too, but a bit more basic - you may want to use some of those for some flavor and use Karatzas and Shreve for other things. $\endgroup$
    – Batman
    Commented Jun 21, 2014 at 5:10
  • $\begingroup$ Any opinions on the book Diffusions, Markov Processes, and Martingales by Rodgers and Williams. How does it compare to Karatzas in terms of completeness and formality? $\endgroup$
    – Jarana
    Commented Jun 21, 2014 at 14:17
  • $\begingroup$ The Rogers and Williams book covers much more ground. They are equivalent in terms of rigor. Rogers and Williams is probably a bit harder to read in terms of leaving out details and the like. Also, Rogers and Williams are a bit more like storytellers, talking about all kinds of stuff, while Karatzas and Shreve is more like a classical textbook. Personally, I always preferred Rogers and Williams, but it's a matter of taste, I guess. If you like stochastic calculus with jumps, He. et al: "Semimartingale theory and stochastic calculus" is great as well. $\endgroup$ Commented Jul 22, 2014 at 6:04

1 Answer 1


I like the book Brownian Motion - An Introduction to Stochastic Processes by René L. Schilling and Lothar Partzsch pretty much. In particular if you are interested in Brownian motion, you will find a lot of interesting stuff about this famous stochastic process in the book (the basics, path properties, construction, the connection to PDEs + Markov processes,..). It also covers stochastic integration (with respect to Brownian motion) and SDEs (existence and uniqueness of solutions, some approaches how to solve SDEs, Stratonovich integral,...). Moreover, there is a solution manual with solutions to (all) exercises.

Very recently, there has been released a second edition of this book. As it contains less misprints and some new material, I recommend to use this one.


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