Assume we have an Ito SDE $$dX_t = f(X_t,t) dt + g(X_t,t) dW_t$$ with $X_t, f(X_t,t) \in \mathbb R^n$, $W_t \in \mathbb R^m$ and $g(X_t,t) \in \mathbb R^{n\times m}$.
If $m>n$, i.e. the noise $W_t$ has more dimensions than $X_t$ itself, then there seems to be some sort of redundancy.
Can I bring the SDE in a form such that $W_t\in \mathbb R^n$ and $g\in \mathbb R^{n\times n}$?
I guess the Idea would be to first define/guess the correct alternative SDE, say $X_t = f(X_t,t)dt + \tilde g(X_t,t) d\tilde W_t$ with $\tilde W_t \in \mathbb R^n$ and then to show that $\tilde g(X_t,t) d\tilde W_t = g(X_t,t) d W_t$, i.e., $$\int_0^T \tilde g(X_t,t) d\tilde W_t = \int_0^T g(X_t,t) d W_t$$
But how would I do that?