I am interested in learning the mathematics to understand and solve stochastic differential equations similar to those seen in quant finance journals.
Currently, the extent of my mathematics knowledge is as follows: Calc II, probability theory, econometrics + time series analysis, and some linear algebra.
I know I will need more calc, ODEs, measure theory/real analysis, and stochastic calc. What is the correct progression of these topics to ensure that I have a strong understanding of the core concepts and which resources should I look at for self study? I realize there may be topics I haven't mentioned, so any additional input into what other topics may be required is very much appreciated.
The study of SDE requires at least a basic knowledge in measure theory and rigourous probability theory. Precisely you should know what is a wiener process and in general you'd need enough theory for understand what is a continuous process. This, more or less, is done in one single book: Introduction to stochastic differential equations, L.C Evans. There is an amazing construction of the Brownian motion, that requires the minimum amount of theory behind.
Edit: obviously you need measure theory and some about real analysis. I think that every book is good because the topic is very standard, but if you have to choose maybe "Analysis" of Lieb, Loss would be fine. Consider that you are talking about a very wide range of notions.