Does anyone have any recommendations for a good book which introduces and cleanly and rigorously explains the measure theory and functional analysis implicit in and relevant to stochastic processes, and then from that knowledge base proceeds to introduce stochastic processes?
I elaborate unnecessarily much below if you don't understand why I would ask for that.
Background: I have tried several times to understand stochastic processes (e.g. Brownian motion, semi-groups of Markov processes, continuous time martingales, SDEs, Levy processes).
Seemingly it should be nothing more complicated than applying measure theory to infinite dimensional function spaces, yet no text seems to take this approach.
I thought I was a very good intuitive thinker, but the approach of most texts, which seem to approach the field through a combination of seemingly unmotivated inequalities and hand-wavy appeals to "probabilistic intuition" instead of trying to explain the conceptual nuances required to understand probability measures/distributions on uncountable spaces, has not worked well for me.
For instance, much of the study of Brownian motion or of stochastic integration seems to rely on calculations using Wiener measure, yet I still have not found a good explanation or definition of Wiener measure (somehow in my class I'm supposed to write proofs using Ito's Formula despite the fact that we never received a precise definition of Wiener measure).
Or when studying semi-groups as related to Markov processes, the resolvent is clearly the "Laplace transform of an operator", but this is rarely if ever pointed out nor ever explicitly/rigorously defined so that I can feel confident performing calculations with the object.
Every textbook I have read so far has had this problem, but the worst offender by far is Feller.
I don't feel like I really "got" probability theory at all until I started to understand it rigorously in terms of measure theory (e.g. Borel-Cantelli, Dynkin's Pi-Lambda theorem, the motivation behind using algebras and then sigma-algebras, random variables as measurable functions, etc.).
But for some reason, despite the fact that the measure theory is much more complicated (as is the analysis) for stochastic processes as compared to probability theory, no one seems to treat it with care, unlike with measure theory for probability. All of the textbook authors seem so excited about the topics that they learned twenty years ago that they just jump into discussions of them without providing the reader the machinery necessary to really understand.
I don't generally understand what it means for a stochastic process to be measurable, since I usually can't figure out what the relevant sigma algebra or measure is supposed to be.
I don't understand the difference between a version or a modification of a stochastic process, or why the finite dimensional distributions don't "determine a process" (since clearly here a more specific definition/idea of stochastic process is implied than the "uncountable set of random variables").
I don't understand how to rigorously define Wiener measure or how to use it for any calculations, especially with regards to optional stopping or stochastic integrals.
I don't understand the idea of a random measure or a probability kernel or a conditional distribution or a regular conditional distribution, or how a regular conditional distribution corresponds to a compact operator (or to a completely continuous operator).
I don't understand how to apply any of my knowledge of functional analysis to the study of stochastic processes, despite the fact that it is obviously very relevant.
I don't understand how to think of Markov processes in terms of operator theory or how to justify that the semigroup and the generator commute, or that the resolvent commutes with the generator, or how or why any operator commutes with another.
The field seems very interesting, and I don't want to give up on it, but right now I feel that I am at an impasse, and hence would greatly appreciate any and all suggestions or ideas for help.