PhD dissertation different from master thesis; What might I be able to do about methodology? I understand that it's possible, thankfully, to have your dissertation in a different topic from your master's thesis, but I seem to have a problem (see below). Do you know any other problems might I run into?

Background on me:

I am finishing up my master's in mathematical finance and now plan to pursue a PhD with my dissertation being in stochastic analysis/calculus, a branch of mathematics used in finance and physics.

My background on stochastic analysis:

I have had 3 classes related to stochastic analysis and 2 classes applying the concepts.

My thesis, being on the P-side of mathematical finance, has extremely little relevance to stochastic analysis, which is more used in the Q-side of mathematical finance.

The ONLY time stochastic analysis ever came up was encountering something in a paper that gives probabilities in the Q-side, and I thought of using a theorem to convert them to the P-side but later found out the paper was unnecessary or inapplicable (I forgot; I think I figured out another solution).

One problem I seem to have:

Some PhD applications seem to require a PhD proposal including a hypothesis, literature review and methodology.

I am not quite sure how mathematical research is conducted. Most of my research has been mainly in finance and statistics. What might I be able to do about this?

I am considering taking a second master's if needed but hopefully it need not come to that.
If you are accepted into a PhD program, you should consider your dissertation topic to be independent of, and separate from your masters thesis topic. A PhD dissertation is fundamentally different from a Masters thesis.  A PhD dissertation is expected to make a new contribution to research (however small), whereas a Masters thesis is a summary and application of existing research.  PhD dissertations are a deeper dive into a topic or research questions, and an attempt to shed new light on the topic or questions.

You may have a Masters thesis that leads naturally into a deeper investigation as a PhD dissertation.  Or not. You should focus your attention on the topics and questions that motivate your interest in a PhD.  What are the core, unanswered questions?  Which of these questions can you, with your skills and interests, address with some chance of success?  

Leave behind the details of your Masters thesis.  Focus on what you have learned in the process, and what new skills or capabilities you have.  Now, with these insights, you should engage with surveys of research in your field -- those describing the unsolved problems or most critical questions.  How does your experience, education, and interests lead you to engage with these unsolved problems or critical questions?