I can give you some recommendations. By the way, I am now pursueing my Ph.D in finance.
1. Real Analysis
A very good elementary textbook is Elementary Analysis by K. A. Ross, which is widely used in math department of USA and European Universities. But for the chinese students, I think this one is a little bit easier. So I suggest you dirctly reading Rudin's Principles of Mathematic Analysis, which can be downloaded from this website; If you have not enough time to cover all the materials of this text and if you have finished Advanced Mathematics in undergraduate, there is a very good book "Real analysis and Probability" by R.M.Dudley or "Real Analysis and Probability" by R. B. Ash. I recommend these two books since they were designed especially for finance students. All the materials in both books are highly relevant with your further study in finance.
2. Stochastic Process
Stochastic Process covers many topics, which can be used in different fields. As far as finance is concerned, I suggest you focusing on Brownian Motion, Jump Process, Martingale. you can find these topics in any standard textbook and in advanced probability. So it is better to study combining with Probability. You can download a Standford University notes by Amir Dembo, which is compact and concise, only 100 pages covering the aforementioned topics. The "Introduction to Stochastic Process" is also a good one without using too much measure theory. This is easier than the former.
3. Stochastic Calculus
If you patiently finished the above materials recommended, the Steele's "Stochastic Caculus and Financial Application" (Springer) is a good candidate. Otherwise you can refer to Elementary Stochastic Calculus by T. Mikosch. this is a readable and of course very simple textbook written for economic and business students. Another very good textbook in this field is Stochastic Differential Equation by B.Oksendal, a classic book.
As for PDE, I have not any good recommendation, because I have not yet spent much time on it.
It is demanding and challenging to read all these books. Anyway if you are interested in finance (taking it as you life-long career) and want to step further in this field, they are prerequisites. Or you can not fully understand modern finance theory and of course miss the fascinating aspects of finance.
I do not know whether you can buy these books in domestic bookstore. If you are in Beijing, you can take advantage of Nation Library. Good luck for your future studying
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