Book request: Mathematical Finance, Stochastic PDEs I'm  a math student, starting a PhD in the near future. My field of research will be mostly in the field of applied mathematics / numerics. Topics will deal with Kinetic Theory, Moment Equations, Fractional Diffusion, Spetral Methods. I think I have a solid background in numerical computing, especially for PDEs. 
Now for my Masterthesis I've dealt with numerical methods for fractional diffusion equations. Since (fractional) diffusion is related to Brownian motion / CTRWs, a lot of authors named mathematical finance as a field that is impacted by their research. The problem is, that I have absolutely no background in finance / econ whatsoever, but I would really like to get into this topic. I think it would open me a lot of opportunities to gain a little expertise in that field. 
I started to read some mathematical finance papers, referenced in the papers I encountered and noticed quite quickly, that I lack the non-mathematical background.
The goal of this question is maybe a bit ambition. I would like to get to know the field of mathematical finance over the next three years. Start with the basics and then move quickly to mathematical finance with a focus on computing / simulation. Since I will do this on top of my work, I would also appreciate books that I can pick up every other day/evening and just read a little.  What interests me espacially are processes that are related to Stochastic DEs, Brownian Motion, Ramdom processes. Especially topics that might rely on the same basics as diffusive processes / kinetic theory. For example, some buzzwords I encountered where stylized facts, options, derivatives (in finance).
Maybe you could split up your recommendations in the categories finance/econ, general mathematical finance and random processes / SDEs in finance.
If you have general remarks regarding my proceedings please feel free to contribute (for example, what do you think about visiting certain lectures offered at my university, broadening my mathematical background, relating my research to the field of finance, software I should get to know like R, etc.)
Thank you very much! 
 A: 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.
A: I have never met a person having a deep background in analysis who struggled to understand finance. It is the other way - having a deep background in analysis trivialises most applications, especially finance. Your question is impossible to answer - you cannot get 'little expertise' in any field. If you really want to learn mathematical finance, then you have to dedicate the majority of your time to learning. It is that simple, there is no shortcut (i.e. asking for book recommendations).
The advice Robert Green gave is excellent, but the book by Revuz-Yor is too difficult for a beginner - I would recommend Stochastic Calculus for Finance, Volume 1 and Volume 2, by Steven Shreve.
A: A great place to get an overview of the field is Investment Science By Luenberger. It just requires calculus and basic statistics. It will help you become familiar with foundational ideas and terminology used across the entire finance industry. Shreve is a great book (in fact wall street firms are know to select interview questions from these books), another is Stochastic Differential Equations by Oksendal. I would consider Risk and Asset Allocation by Meucci a pretty indispensable item for the mathematical finance bookshelf. Monte Carlo Methods in Financial Engineering by Glasserman contains a lot of theory and techniques which become important in many financial models, especially option pricing. To be good at mathematical finance you really need to master two subjects: statistics and linear algebra. I found in our PhD program that students who did this really excelled. For the statistics side I would concentrate on multivariate statistics and time series. For the linear algebra side I would move in the direction of matrix analysis and eventually functional analysis.
