Current Stats PhD Student with evolving research interests

[mathsimp]

Long time lurker here - I'm currently at a mid ranked PhD program in statistics and just about to start my second semester of my first year (wrote my apps in 2020 for admission for fall 2021).

Between the time I was admitted and now, I feel I am more interested in more theoretical and abstract mathematics such as measure theory, functional analysis, topology, probability theory, and stochastic processes - as opposed to machine learning, data science with biological applications (which were my original interests that led me to statistics).

Our department has maybe about 2 faculty, out of 10 or so who seem to produce students who do work in the area remotely related to what I am interested in. However, in our (applied and pure) math department has maybe 5 faculty publishing in this area in journals such as Annals of Probability and of the such. I suppose having a few of them on my committee would be a good idea, however I have been told that these faculty are swamped with their own students and is unlikely to take students from outside their department. Despite all this, my end goal is to be in quantitative research, probably in the industry as opposed to academia.

I'm concerned that I may not be able to do my research I am interested in my own department. There are three routes in my opinion, in the order of least to most time consuming:

1) Go through with the statistics PhD and try to align my research as close to what I originally hope it to be
2) Reapply to PhD programs that have a stronger emphasis on the mathematical side.
3) Drop out, get a masters in (applied)math for the letters, and apply for applied math or pure math programs

Is there anything else I can do to be able to study more math, specifically stochastic analysis as a culmination of topology, measure theory, and functional analysis? Perhaps staying in a statistics program can help me apply stochastic processes to machine learning algorithms or population biology - which would be interesting and we have faculty working in this area, but I'm hoping the meat of my research is still going to be more about the math behind it.

My profile for reference:
 

BA: Economics @ Ivy League (CGPA: 3.5)
Relevant courses: Calc 1,2,3 (A,A,A+), Linear Algebra (A-), Probability (A), Inference (A-), Mathstat (A), Regression (A), Real Analysis I, II (A, A-) - included measure theory

MS: Statistics @ Large State School (ranked 35-45) (CGPA: 4.0)

PhD Statistics @ Mid ranked department (ranked 40-50) (CGPA 4.0 - does this really even matter at this point though?)

Any advice from current faculty or grad students is greatly appreciated.

[bayessays]

If you want to end up in industry, it doesn't seem to make sense to switch departments and do something more theoretical.  You say your current department has 2 faculty doing stuff you're interested in - why not work with them?  You only have to have one advisor, so it shouldn't matter whether your department has 1 faculty or 5 in the field.  If your current department is making you miserable and you can't imagine continuing the research, then I'd consider transferring or mastering out, but I'd focus on whether switching will actually help you accomplish your goals.  You can always learn more about these topics on your own, or switch directions during a post-doc, etc.  Switching PhD programs is a big step that involves going through the first year or two (the worst part for many people, or at least the busiest in terms of classes/exams) all over again, which is a total waste of time.