Before asking this question, I have read this meta and I believe I have a few specific requirements that will reduce the chances of this question to be opinion based.
I am looking for a book on differential equations. To set some goal, let's say I want to be able to understand what Black–Scholes model is about and how it works(from the mathematical standpoint).
As for my current education, I have a PhD degree in applied mathematics and spent several years working in Machine Learning domain in industry(the industry I'm working in is not related to finance). I have never had a course on differential equations though.
Taking into account my current background, any suggestions to get me started ?
Thanks
A good book that I used during my first semester of graduate school is An Introduction to Financial Option Valuation: Mathematics, Stochastics and Computation by Desmond Higham. Professor Higham maintains a website for the book which includes Matlab code and hints to the odd number exercises. No knowledge of measure theory is necessary and almost all of the derivations are done through techniques learned in calculus and differential equations. The Black-Scholes PDE is covered in the second half of the book.
Another good supplementary book is Financial Calculus: An Introduction to Derivative Pricing by Martin Baxter. This book goes through Ito’s Lemma before the derivation of the Black-Scholes PDE.