While defining the Ito integral, we generally take forward difference. Then we go on to prove many properties, one of them being Ito rule. Formally, we could write the Ito rule as $$ df(X_t) = f'(X_t)dX_t + \frac{1}{2} f''(X_t) d\langle X,X\rangle_t $$ where $\langle X,X\rangle_t$ is the quadratic variation of $X_t$ and for Brownian motion $B$ it is simply $\langle B,B\rangle_t = t$.
However, if we now define the Ito integral using backward difference, will the Ito rule as stated above still hold? And will the quadratic variation of the Brownian motion still be $$\langle B,B\rangle_t = t$$ or would it be $$\langle B,B\rangle_t = t=-t$$ now?
Could you also provide any tutorials or references which would cover these topics so I can read and understand them properly?
Thank you!
Note: I know that my question may not be in a very precise mathematical language, and that it can be written in a much more rigorous way. However, I am more interested in understanding the broad ideas, which I believe can be expressed compactly in the notation I used.