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Which books on financial mathematics would you recommend for people with good background in probability, statistics and stochastic processes but without any background in financial mathematics?

The goal is the next one: to find interesting mathematical problems in financial mathematics, solve it (using methods of probability, statistics and stochastic processes) and publish an article about it. So the question is related to scientific work, not with industry. And the question is not about algorithms or something like that.

If my question is too general then which books on main branches of financial mathematics would you recommend?

The only books I've heard of is books of https://en.wikipedia.org/wiki/Mark_S._Joshi.

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    $\begingroup$ It is a very general question, and depends on the sort of quantitative finance you want to do. For example - if you wanted to do algorithmic trading or alpha research, I would refer you to econometrics textbooks. If you wanted to do derivatives pricing I would indeed send you to one of prof Joshi's books or the shreve books. Also depends on if you want to work in industry or not - if you are working in industry you will need to learn how to program well as it is all you will be doing all day. $\endgroup$
    – rubikscube09
    Feb 10 at 20:44
  • $\begingroup$ @rubikscube09 Thanks for helping to make the question more specific. I changed the text of the question, could you please see a new version? $\endgroup$
    – Botnakov N.
    Feb 10 at 21:15
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    $\begingroup$ I think your question is a duplicate. But I also think you're approaching it from the wrong end. I suggest that you look at what people publish, trying to solve open practical problems - random example 1: risk.net/cutting-edge/views/7956005/… discussing some work by Fabrizio Anfuso who disgrees with some recent work by Andy Dickinson and by Michael Pykhtin. Example 2: look at the recent articles by the later Peter Carr scholar.google.com/… . As you can see, $\endgroup$
    – Dimitri Vulis
    Feb 11 at 16:24
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    $\begingroup$ most open interesting/ractical problems seem to be NOT "how do I price this", but rather "how do I hedge this" or "how will the price change if these inputs / assumptions change" or "how do I characterize this property using some pretty math". Once you see what problems you'd like to solve, you'll be in a better position to see what mathematical machinery, if any, you need to brush up on. And then you can look for books on this machinery, maybe. $\endgroup$
    – Dimitri Vulis
    Feb 11 at 16:29
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    $\begingroup$ A user of this forum, Dr. Dave Harris, has proposed some interesting, novel methods for asset pricing utilizing a Bayesian framework. His working papers are here. $\endgroup$
    – amdopt
    Jun 5 at 9:48

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I think it is difficult to come up with an open problem in math finance that is also economically/socially relevant. And I don't mean to say "practitioner" relevant, as you mentioned you are looking for a topic for your academic research. Granted, derivatives are more mathematical than underlyings, and are disproportionately represented math fin books. But I think there are also good fundamental questions in stock and bond markets that require a better math than money-making trading algos. I recently discussed, and more posed than solved, one of such problems related to flow networks describing cyclic and acyclic flow of money and securities in a broad financial market: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4473754. If you are an Ito formula person, this is probably not your thing. But any contribution to understanding how the topology of money flows affect efficiency and elasticity of the markets would probably be of broad interest, including for regulatory consideration.

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