Coordinate change of the solution of an SDE

Question

Let $X_t$ be the solution to an $d-$dimensional SDE $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,$$ and let $A$ be an invertible $d \times d$ matrix. Let $Y_t=AX_t$. Which SDE does $Y$ solve? I feel like it is like a coordinate change. The Ito's formula does not work here because the transformation is multivariate. Does anyone have an idea?

Answer 1

Obviously, $dY_t=A\,dX_t$ and $$ A\,dX_t=A\,b(A^{-1}Y_t)\,dt+A\,\sigma(A^{-1}Y_t)\,dB_t\,. $$ Setting $\tilde b(y):=A\,b(A^{-1}y)$ and $\tilde \sigma:=A\sigma(A^{-1}y)$ gives the SDE of $Y_t$: $$ dY_t=\tilde b(Y_t)\,dt+\tilde\sigma(Y_t)\,dB_t\,. $$