# Reference on Stochastic Integration with Measure Theory but little to no Real Analysis

Similar questions have already been asked, for example here and here, but my situation is a bit specific so that is why I am asking. I am interested in a book that presents rigorously and with proofs the theory of Stochastic Integration/Stochastic Calculus.

• I already have a good understanding of the matter through advanced applied mathematics classes, application-oriented books such as Stochastic Calculus for Finance II: Continuous-Time Models (Shreve) or Elementary Stochastic Calculus with Finance in View (Mikosch) as well as internet sources here and there, so what I am seeking is to strengthen my technical understanding of the field (i.e. "review" it from a more rigorous and formal perspective);

• I have a background in Measure Theory which is primarily based on the following notes (in French), essentially up to chapter 10 included: Théorie de la Mesure et Intégration $-$ so a good understanding of measures and of the construction of the Lebesgue integral, including the main convergence theorems.

• I have a strong foundation on Probability and a solid background on "basic" Real Analysis but I haven't thoroughly studied more abstract Real Analysis concepts, for example spaces. I am more or less illiterate in Topology $-$ unsure whether this matters for Stochastic Integration Theory.

To put it on another way, it seems to me that most Stochastic Integration books assume a good understanding of Real Analysis and/or Measure Theory and provide the reader with a thorough introduction/appendix to Probability, but I would need the inverse.

Based on my own experience studying Measure Theory as well as other sources such as the answer to this question or this other one, I have the impression it is possible to develop the theory of Stochastic Integration relying almost entirely on Measure Theory and very little on Real Analysis.

Hence I would like:

1. A proof-based book on Stochastic Integration which 1) stands on Measure Theory but 2) avoids advanced Real Analysis (e.g. Hilbert or Banach spaces, etc.) and Topology or keeps them to a minimum, as I am less familiar with those areas. The book should be rigorous and present proofs to theorems (but avoid getting too technical à la française).
2. Additionally, a "refresher" on essential Measure Theory machinery for Stochastic Integration wouldn't hurt me, so it would be helpful if the book contains some preliminaries or a sizeable appendix on Measure Theory. If you know a good book $A$ that fulfills point 1 above but without a Measure Theory refresher, and a book $B$ with a good introduction or refresher to this topic, please list it too.
3. As a add-on, if the book treats the topic of martingales and stopping times, it would also be helpful.

Some references I believe can fulfill these conditions are the following:

As an example of what I want to avoid is Brownian Motion and Stochastic Calculus (Karatzas and Shreve) which is too technical and formal.

Do you have any additional suggestions on books? What about those listed above?

• Can the down voter kindly explain? – Morris Fletcher Nov 3 '17 at 11:42
• I would suggest "Brownian Motion - An Introduction to Stochastic Processes" by Schilling & Partzsch in combination with "Measures, integrals and martingales" by Schilling (if you need additional stuff on measure theory, martingales, stopping times, ...). Once you have mastered stochastic integration with respect to Brownian motion, you won't have any problems to understand stochastic integration with respect to martingales. – saz Nov 3 '17 at 12:12
• Thank you for the references @saz. Measures, Integrals and Martingales seems particularly fitted as a refresher. I have peaked in and in the beginning the author states that "Undergraduate calculus and an introductory course on rigorous analysis in $\mathbb{R}$ are the only essential prerequisites" : in your opinion what concepts from Real Analysis does the author think are essential? – Morris Fletcher Nov 3 '17 at 13:07
• Why not simply give it a try? Since I'm not the author of the book, it is kind of hard to summarize for me what the author means by "introductory course on rigorous analysis". – saz Nov 3 '17 at 14:59
• Fair enough @saz, I'll do that, that reference seems really great anyway! If it is not too much to ask, what is your opinion on Protter and Chung & Williams $-$ i.e. combining them with Schilling's M.I.M.? I have read than Protter treats the subject in a "novel" way (or at least novel when it was written) that allows him to be quicker and less technical. – Morris Fletcher Nov 3 '17 at 15:15