I am a graduate student and I want to learn stochastic calculus by myself. I have learned measure theory, real analysis, point-set topology and advanced probability theory (David Williams, Probability with Martingales and Kai Lai Chung, A course in probability theory, such as independence, LLN, CLT, discrete-time martingales) before.
The topics that I want to learn include: continuous-time martingale, Brownian Motion, Stochastic Integration, Stochastic Differential Equations, and the general theory on semimartingale.
Now I have seen many answers on the textbooks and I decide to choose books from the following four classics:
Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus
Daniel Revuz and Marc Yor, Continuous Martingales and Brownian Motion
Rogers and Williams, Diffusions, Markov Processes, and Martingales, Vol 1&2
Philip E. Protter, Stochastic Integration and Differential Equations
Since time is limited, I cannot learn all these four books. My question is that: what are the differences between these books and which one should I use? Maybe there is no book that covers all the contents, which book should I refer to for each topic?
Many thanks in advance.