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I've studied physics at a university previously and actually earned a degree in theoretical physics, but then switched over to mathematics, where I focused on stochastic analysis/calculus/processes (I'll just call it stochastics).

Now, I remember taking a course on stochastics while studying physics, and later I've had several courses on stochastics while studying mathematics. The topic seems to be the same, but the approaches are vastly different.

In mathematics, as a basis, you have measure-theoretic probability, Lebesgue integration, different types of convergences of random variables, some functional analysis, Brownian motion, Ito integral, Ito lemma, stochastic (Malliavin) derivatives and so on. These are the topics I've studied in some detail (wrote a master's thesis on Malliavin calculus)

So recently, I came across my (very) old notes on stochastics from my physics studies, and, suprisingly, it all felt foreign to me. The approach taken there is somewhat strange:

1. Expected values are defined as averages of r.v. ensembles (or something -- there is no formal definition of a r.v. or ensemble)

2. The is absolutely no notion of a probability space (sample space, sigma-field, measure, filtration -- nothing)

3. No mentioning of Ito integration, yet there is a great deal of SDEs, which are all written in a form requiring some notion of a white noise process.

4. Delta function is all over the place (anyone else with more mathematical background annoyed by this when reading physics literature?)

While I understand that it is not always necessary to be mathematically rigorous in physics, still I'm wondering, do you think this was just not a very good course, or is there truly an alternative (and rigorous) setting to work with SDEs, without measure-theoretic probability, I'm not aware of?

Now more importantly, there were also some topics discussed in these lecture notes, that I have never seen discussed in any "mathematical" book on stochastics, i.e. in a more traditional framework (measure theory, probability spaces etc):

1. Kramers-Moyal expansion

2. Fokker-Plank equations (ok, this one is mentioned sometimes, but still in a somewhat different form)

Is this (especially the first point) something that simply has a different name in a "mathematical" approach to stochastics? Can you suggest any books, that explain these topics in a traditional mathematical setting?

Why is it so that Ito calculus is a fundamental notion required to study stochastics for mathematicians, but there is often no mentioning of it in stochastics for physics? Why is anything like Kramers-Moyal expansion never discussed in courses on stochastics for mathematicians? I'm not even sure how one would go about to introduce the related results in a measure theory + Ito calculus approach.

Sorry for not being able to form a more concrete/exact question. This comes from my confusion on the subject.

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# Stochastic calculus in mathematician's vs physicist's view

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