Q:

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!

