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I am looking for a book which teaches mathematical topics which are relevant to master quantitative finance and statistics.

Please note, I do not mean a book which would explain how math is applied to solve problems in quantitative finance and statistics but rather a book which would teach mathematical toolkit (linear algebra, set theory, probability, stochastic calculus, measure theory, etc.) needed to further understand math applications in the above mentioned fields.

One example is this book, but I need more.

The book should:

  1. cover only those topics in math which are relevant for quant finance ans statistics;
  2. be self-contained or make clear what previous knowledge of math it relies upon;
  3. rigorous (no "just believe that it is true" or "we present an intuition instead of a proof");
  4. provide exercises with available solutions;
  5. be a series mathematical book (proofs are must).

Thanks!

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All the topics you've mentioned are wonderful and shouldn't be eschewed by reading some finance-oriented review book. I recommend these instead.

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  • $\begingroup$ Another vote for Halmos. $\endgroup$ – Nathan S. Mar 17 '15 at 3:08
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I would recommend the books from Steven Shreve. Here is a link to some one of his older online pdf's (1997 but nevertheless true) so you can check if that fits the bill. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.137.6951&rep=rep1&type=pdf

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