I just recently decided that I'd like to pursue a PhD degree, which means I did not do any research previously. I am currently in a MS program and doing research now. I will graduate at Dec 2022 but plan to have a gap year to do research.

Undergraduate institue: UC with math rank ~20. Major math applied

GPA: 3.81 overall, 3.95 for math

No research

Graduate institue: Top Private but well-known for cash-cow masters lol. Major: Operations Research. Taking PhD courses and doing research now. Hope that this will help! PS: This private school should be the only one that differentiate PhD courses and masters courses lol.

GPA: N/A

GRE General: V162(90%) Q170(96%) AW 4.0 (54% I think)

GRE Sub Math: Did not take, I do not think this will help a lot, but just assume I got a good score like 95% or 90%?

Field of Interest: Probability and Stochastic Process, Machine Learning, Mathematical Finance (though this is actually an application of stochastic process and ML)

Should I apply to math PhD focus on Probability or Stats PhD? I am actually not that interested in general statistics (something like estmations, inference, and hypothsis test...) Indeed Stochastic process is my primary interest, and I am interested even more in analysis rather than statstics. However I am not confident to be Pure Math PhD and will consider some applications such as math finance.

Math or Statistics? Should I continue to take PhD courses (will this be helpful? PhD courses definitely take a lot of time and I am not confident to get A. For non-PhD level courses, it is much easier and I will get A with little efforts.) What kind of schools shold I target at? I was considering CMU mathematical science, not sure the selectivity, but this contains probability and mathematical finance research and it is probability easier to do research on Machine Learning there?