Stochastic calculus book recommendation I'm a quantitative researcher at a financial company. I have a PhD in math, but I'm an algebraist, so I only took the two required analysis courses in grad school (measure theory for the first, and I don't even remember the content of the second course. It involved Fourier series).
I taught a probability course four years ago. It was brand new to me; I learned it as I taught it. I forgot a lot of it, but I was able to pass the first actuarial exam last year. We didn't do stochastic processes. I know the definition but have done virtually nothing with them. 
I want a book from which to learn stochastic calculus and be able to apply it to my job (financial modeling). Being a mathematician, I don't need the book to hold my hand, but beyond undergraduate probability the book would preferably be self contained. Does anyone know of any appropriate books?
 A: Stochastic Calculus and Financial Applications by J. Michael Steele is the book for you, in my view.  This is definitely an applied math book, but also rigorous.  The author always keeps finance uses in mind although building concepts from the ground up.  Some consider this "hard" but that's because they may not have the math training for it (some may never have done a proper proof, for instance). This will certainly not be a worry for you. Finally, the book is enjoyable and reveals the beauty of the subject. 
I have also used Baxter and Rennie, but I found it a little painful.  This book is written so that it is accessible to people who have done not more that 2 years of college math (it's enough if you know some calculus and other basics that you might have learned in high school or freshman year).  That's great if you are in that boat, but it contorts itself to make that possible.  If you have sone some measure theory even a while ago, you are no longer in that boat. 
Here is another one by Shreve. Also great. But no one focuses solely on applications as sharply as Steele does without ever sacrificing rigor. 
Then there are various specialized topics like stochastic volatility with good books, but that that's not a starting point. (Although, the first 2 chapters of this book are an interestingly different presentation of general stuff). You will also find a bunch of good books on option pricing.  Course pdf on stochastic Calculus for finance and aplenty on google. Do look to see what you may like. 
This book on Stochastic Calculus by Karatzas and Shreve is also great and many have gone to the industry with this as part of their training but 
perhaps leans too theoretical for your needs and is not specifically for finance. 
My background in the area at that time was pretty much what you have now. Taught myself “streetsmart” probability when I decided on the career move and went from there. Please note, your probability questions in interviews (as opposed to the job) will usually not involve any measure-theoretic probability, just puzzles and problems that you can find in puzzle books and even here on MSE. But you are already in a job and know that. 
A: I recommend starting with:


*

*Baxter and Rennie : http://www.amazon.co.uk/Financial-Calculus-Introduction-Derivative-Pricing/dp/0521552893 and

*Björk : http://www.amazon.co.uk/Arbitrage-Theory-Continuous-Oxford-Finance/dp/019957474X
I recommend them because I like the intuitive explanations they both provide in a first contact with stochastic calculus related to finance. Moreover, they both provide "solvable" exercises for practice. With "solvable" I mean that they are generally closely connected to what the level of the main text.  Baxter and Rennie, also has available solutions in the book. 
I have not compared them with any other book. 
Last, both of them can be easily obtained online.
For stochastic calculus you can easily find more hardcore stuff with google. 
A: I think that the books by Karatzas and Shrieve "Brownian Motion and Stochastic Calculus" should serve you well since it definitely doesn't hold your hand but introduces things in enough rigorous detail to be very profitable for someone with a good mathematical background (and it is less chatty than the book by Rogers and Williams or simmilar books, while being much more approachable than books like Revuz and Yor).
Another perk is that it is the foundation for their "Methods of Mathematical Finance" which is one of the more rigorous texts on math finance, so probably what you're looking for.
