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:
- 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).
- 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.
- 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:
- Stochastic Integration and Differential Equations (Protter) $-$ I have checked the 1st chapter and seems very close to the style I am seeking.
- Stochastic Differential Equations (Oksendal)
- Introduction to Stochastic Integration (Chung & Williams)
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?