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Mathematical finance research?

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Yes, mathematical finance is an active area of research, and there are journals (and scholars!) dedicated to math finance.   


As for the PhD program, it depends on the institution that you are thinking about. Math finance scholars can be found in math departments, but also applied math, statistics, operations research, financial engineering, finance and economics.
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Math finance is a pretty big area for research.

>Are there any people out there who are not quants in industry, but do research in statistical or mathematical finance and publish?

Tons. Many of the financial mathematicians of today were the analysts/probabilists of yesterday (eg. Karatzas of Columbia).

You can also approach a lot of the definitions in stochastic analysis from a financial viewpoint, and finance motivates quite a bit of modern research in stochastic analysis or other fields (eg. Banach lattices).
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There is such a field called mathematical finance. However I do want to warn you that depending on your expectations of the field you might be disappointed. It's rare these days for results from academia to be useful in industry and likewise mathematical finance isn't math so it's a bit alienated from the rest of probability as well.

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