It seems that there are researchers in probability in both stat and math programs. What's the difference really between being a probabilist coming from a stat department as opposed to a math department? Can you study SDE's in a stat department?

You might find that stats programs are a bit more applied and tend to be more involved with interdisciplinary research. Depends on the school though!

From an applicant's p.o.v. - most math programs required the subject gre, whereas stats programs tend to only require the general gre. (This is not a scientific survey, just from looking at department pages.)

From an applicant's p.o.v. - most math programs required the subject gre, whereas stats programs tend to only require the general gre. (This is not a scientific survey, just from looking at department pages.)

Then if someone wanted to study mathematical finance and SDE's which would be better: math or stat dept?

Your best bet would be to find programs which offer than combination of research interests, and go from there.

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