Further Reading on Stochastic Calculus/Analysis 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.
 A: just to offer two cents on this (health warning is that this is from an ex-trader rather than quant, and based on personal experience through self-study rather than eg as part of a formal PhD programme):
Williams really is fantastic, learned the basics of measure theoretic probability from that as an undergrad, and it's stood the test of time and is still a classic.
After this, some natural canonical texts would be:


*

*Rogers and Williams' two volumes: Diffusions, Markov Processes and Martingales


I think this really does contain a huge amount of material (in addition to containing the material from Probability with Martingales so in some sense is a natural transition). You'll certainly find the standard Ito/Girsanov/Radon-Nikodym material well presented therein.
A different tack would be:


*Oksendal: Stochastic Differential Equations, 


This was very popular when I was reading up on SDEs, and has a somewhat less formal style than some of the other standard references.


*Karatzas and Shreve: Brownian Motion and Stochastic Calculus


This is a bit more encyclopaedic than Oksendal, but again was very popular when I was reading the material about 10 or so years ago. More heavy going than Oksendal, and possibly overkill if the ultimate aim is more finance than analysis orientated.


*Revuz and Yor: Continuous Martingales and Brownian Motion


I didn't use this myself, (again I was reading for interest and as ancillary to finance rather than for embarking on a stochastic analysis PhD - am a number-theory/algebra nut at heart!), but this is I think a classic text, although more formal than the others I've mentioned.
Finally a rather pleasing book is:


*Bobrowski: Functional Analysis for Probability and Stochastic Processes


This has a nice survey, as the title suggests, of some of the functional analytic underpinnings of measure-theoretic probability, and I found the exposition a delight to read.
Hope some of those help, these are not finance books (sounds like you've got that covered), definitely can offer some views on that side of things should you need depending on which area of finance might interest most (credit/rates etc..). Many of the finance books by authors such as Brigo are highly rigorous but much much better suited to assimilating the finance concepts and acquiring facility with actual problems that matter 'at the coal face', but again depends on the aim / perspective.
Good luck and cheers!
A: 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.
