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https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf

In the above document stochastic taylor expansions are nicely explained.

Let us now consider a typical SDE model in finance like SABR. here the form is:

$$ dS_t=b(\sigma_t,S_t)dW_t $$

My point is that because $b$ also depends on another stochastic variable, we cannot follow the simple steps provided in the document. Or can we ....... So my question, how dos the Taylor expansion of $S_t$ look like in this setup? To keep it simple we can just say first order, that is not not important.

And also: can we even derive the Taylor expansion if we dont know the dynamics of $\sigma_t$. I wonder if we can just say $$dS_t=\sigma_t b(S_t)$$ and treat $\sigma_t$ as a "constant". How wrong would that be?

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