TITLE: Further Reading on Stochastic Calculus/Analysis
QUESTION [7 upvotes]: I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal.
So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on probability with measure theory and some courses applying those concepts.
The probability reference was some chapters of Probability with Martingales by David Williams.
Where do I go from here? What textbooks do you recommend?
I believe I lack knowledge on a lot of the basics such as, different types of convergence of random variables, laws of large numbers, Malliavin Calculus (and the calculus of variations), Radon-Nikodym stuff, proofs of basic stochastic calculus results (like Girsanov theorem and Ito's lemma), etc.
I'm also looking up recent journal publications. Know where I can look?
Please provide feedback if you think the question can be improved. Any help is appreciated. I don't mind anyone posting comments as answers so I can upvote, I guess.

REPLY [2 votes]: I think it takes courage to identify what you are weak at and make that public. The mindset of identify weaknesses and seeking to improve is essential. If you is able to keep this mindset awake even when struggling, progress will be but a mere trivial corollary to the rich understanding that you will gain.
Here are some thoughts, based on my experience:
a) I remember (painful memories are usually recalled quickest) when I didn't know what a Radon-Nikodym derivative was. Nor did I really understand why we care to study martingales. At times studying this stuff was (and still can be) torture.
Fortunately, I now do understand. How did I get here? By studying and really trying to understand what the authors are saying. Discussions with fellow interested peers helped. Computer simulations helped. 
b) Stochastic analysis is a subset of probability theory, a field of mathematics created to satisfy gamblers. If you are into gambling, stochastic analysis will make sense. Physicists also love stochastic calculus, but for different reasons.
For example, topics such as 'hitting times' will be obvious to any (good) gambler - the gambler wants to know what level he should hit before he can leave the casino.
There are a lot of good books, non mathematical nature, that give context on the history of stochastic analysis and how it came to dominate finance. Physicists on Wall Street by some bloke (a physicist) is pretty good. 
c) The books listed by MathVandal are fantastic. Williams's book is superb: at the start he has the (successful) candour to explain the purpose of measure theory and topological spaces in a few lines. There are many other books worth reading: Schilling has a book on martingales and measure theory, Kreyszig has a book on functional analysis and Klebaner has a book on stochastic calculus. I despise many mathematical finance books, but anything by Wilmott is pretty good.
d) If I were you, I would not be punishing myself if you do not understand the Malliavin calculus. It is a very difficult topic. Yes, it is just the calculus of variations applied to stochastic processes on some space - that is trivial, when you think about it. But Malliavin's idea is hard to express - there are many spaces that are degenerate in some sense and Malliavin invents new tools that attempt to tackle these degeneracies. For example, he invents 'quasi surely' as opposed to almost surely to refine behaviour of paths. It takes a lot of time to understand this as opposed to the plethora of symbols and theorems that are thrown at you.