Applied Math Foundational Books I have a BA in mathematics from a pretty good school, where I effectively exhausted the mathematics sequence.  The sequence mostly focused on pure math (including measure theoretic real analysis and some functional analysis).  I am now interested in learning about financial mathematics, at an intermediate to advanced level.
I have studied probability to what I would call an "intermediate" level (using Feller, Ross's "Introduction to Probability Models", Gut's "An Intermediate Course in Probability", and I have begun learning probability in a measure-theoretic context.  I probably have some gaps in my knowledge here, for example, with branching, recurring processes, and queues, and ....
I am, however, concerned that this is not enough of a foundation in applied mathematics.  For example, I have had very little exposure to differential equations (and I couldn't solve one to save my life).  I have seen references to stochastic differential equations, so obviously, I should learn about the deterministic kind first.
Can you suggest a list of applied mathematics books that would serve as a foundation for an advanced course in applied mathematics?  What I mean is, I suspect I will have to fill in the gaps in my knowledge.  What books would you suggest as references/introductions to the field?
 A: Well, if you're serious about applied mathematics-and serious in that you don't just want "reciepe" books,rather applications that build on the meaty theory background you have-then you should avoid such texts and try and locate books that don't avoid theory,but merely downplay it. Those are the "real" applied mathematics textbooks. 
You definitely need to strengthen your background in differential equations, you're pretty much dead in the water without that in your background. The best beginning textbook I know on differential equations is George Simmon's Differential Equations with Applications and Historical Notes.  It is one of the most beautiful, richest textbooks you've ever find on any subject-it covers all the basics of ordinary and partial differential equations using only basic calculus and a context of not only physical applications,but incredibly detailed and scholarly notes on the historical founders of the subject. It's a must have for any mathematical library. The one flaw the book has is that it doesn't explicitly use linear algebra,but rather old fashioned linear equation notation. But by all means,don't let that chase you away from one of the best books there is on any subject. It really is the book to start with. 
Another subject you'll need to be very comfortable with in terms of applications is linear algebra and there's no better book from an applied standpoint for this then Gilbert Strang's classic Linear Algebra With Applications. Be careful with this because there are several versions of this book and the one you want is this one-the other one is much gentler and less substantial. This is the one with the real red meat by one of the great applied mathematicians of our era-with a host of applications you won't find anywhere else,both standard and exotic. While you're at it, check out Strang's calculus text online at his website. It may be the best basic calculus text there is bar none-with more applications then just about any other text in it's wieght class. And best of all, it's free!   
For financial mathematics,you really need a good background in probability theory. It sounds like you have a good working knowledge of the brute theory, but that'll only take you so far. You need to back up and learn some of the basics first, particularly the applications. One of the best books that currently exists on this subject for the beginner, which contains a host of applications, is Grinstead and Snell's Introduction to Probability. It provides not only a comprehensive undergraduate course in the subject,but it provides many applications, particularly to finance and the social sciences. Best of all,it's available online for free for download. 
Lastly. a very good addition to your training will be advanced calculus with applications. Not "applied advanced calculus",which is mostly receipes in metric spaces. I'm talking about a relatively recent and very important trend in elementary analysis textbooks where real-life applications are included alongside a careful presentation of the theory of calculus. This is where the artificial separation of pure and applied mathematics-which is there for purely profiteering reasons in academia in my opinion-is taken down and both are discussed in an advanced calculus course in equal measure and import. There aren't many of these books yet, sadly-but the ones that do exist are excellent.The best one that currently exists is Jeffery Cooper's Working Analysis,in which a thorough course in advanced calculus of one and several variables is peppered with applications to physics, biology, numerical analysis, economics and so much more. This is a terrific text that most students of mathematics should have in thier library. A bit less advanced but also quite good is Donald Estep's Practical Analysis of One Variable. This is really an honors calculus book masquerading as an analysis book, but Estep writes beautifully and presents one variable analysis in complete unity with literally hundreds of applications, many fascinating insights and an interestingly original presentation. This one is well worth having as well,particularly for you since it's emphasis is on differential equations.Lastly and more sophisticated then either of the previous texts is Real Analysis and Applications: Theory in Practice by Kenneth R. Davidson and Allan P. Donsig.This isn't really a full blown analysis course but rather a supplementary text following up a standard intermediate analysis course based on baby Rudin or Charles Chapman Pugh's book. That being said,it has an amazing set of applications, including Fourier analysis and wavelet theory. It's definitely worth a look.  
That should get you started. I'm sure the others will recommend other good texts as well and if I think of any others,I'll edit this post and add them. Good luck! 
A: I recommend Gilbert Strang's books: 1) Introduction to Applied Math; 2) Computational Science and Engineering.
