If we have an a process $X_t$ with values in $\mathbb{R}^{n \times n}$ which solves a linear Stratonovich SDE $ dX_t = A_t X_t dt + B_t X_t \circ dW_t $ then the inverse of $X_t$ exists and solves $ dZ_t = - Z_t A_t dt - Z_t B_t \circ dW_t $ It is easy to see that $X_tZ_t =$ Id by using the product rule. My question is this:
If $X$ instead solves the affine SDE $ dX_t = (A_t X_t + a_t) dt + (B_t X_t + b_t) \circ dW_t $ does it have an inverse?