Non-commutative Ito Formula Does there exist a formula of the Ito lemma for matrix valued processes under matrix multiplication?  
That is where
$$
\Delta X_{t+\Delta t} \neq X_{t+\Delta t} - X_t
$$
but instead
$$
\Delta X_t = X_{t+\Delta t}X_t^{-1}?
$$
 A: To begin with, if we stick to semimartingales, one assumes these have right-continuous paths with left limits. Correspondingly, if $X_-$ denotes the process whose values are these left limits (so necessarily $X_-$ is predictable) one may define a new semimartingale by setting $\Delta X := X-X_-$. Note that $X_-$ itself may not be a semimartingale because it is no longer necessarily right-continuous.
To get back to your question, if $X$ and $X_-$ are different from zero then one may define a new process, typically denoted $\mathscr{L}(X)$ and called stochastic logarithm, by setting
$$\mathscr{L}(X)_t:=\int_0^t \frac{1}{X_{s-}}dX_s.$$
For this process one has the equality
$$\Delta \mathscr{L}(X) = \frac{\Delta X}{X_-},$$
which formally resembles your expression if you identify $t+\Delta t$ with $t$ and change $t$ to $t-$. 
This definitely works in one dimension. I suppose that if the object $X_-^{-1}$ is a well-defined locally bounded matrix-valued process the same operation (stochastic logarithm) can be performed in higher dimension. In such case standard multivariate Ito lemma would be applicable to functions of $X$.
