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
Best Answer
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.