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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 $$ \Delta X_t = X_{t+\Delta t}X_t^{-1}? $$

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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 $X$.

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