You could start with the first chapter of this book, and then with this three-volumes book. The former is a very nice mathematical introduction to finance, from the viewpoint of someone on the mathematical (or physical) side. The latter may seem, and is, a book on interest rates, but it allows you to cover all mathematical techniques used in finance nowadays, and its first volume is the best introduction I have ever seen on mathematical finance ; it has btw a very nice bibliography that will redirect you to central papers in the discipline etc. I am not that fan of this book, even if I started in the field with him, but it could be ok nevertheless for what you are looking for. Finally, there is a book that is not very good on mathematical finance at all, but it is central on FX implied volatility quoting conventions, and is a must have for this.
Last point, previous books are not books on stochastic processes or PDE's or other mathematical subjects that are used in mathematical finance, they are books on mathematical finance roughly covering these subjects, and using and applying them to fianance - essentially pricing and hedging, curve building etc. This means that sometimes you will need to put your nose in a book or another on stochastic processes or even probabilities (note that this book on probabilities and discrete time martingales is a must-have), or PDE's etc. Theses are my complementary advises. I know that this wasn't you primary question, but I don't see myself giving a piece of advise on mathematical finance without mentioning this.
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!
Best Answer
Maybe this answer is a bit late for you but I am sure there are a lot of other people with the same question. To them, I recommend the following: