TITLE: Book request: Mathematical Finance, Stochastic PDEs
QUESTION [9 upvotes]: 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!

REPLY [2 votes]: 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.