Let $W$ be a two-dimensional Brownian motion. Let $A^a$ and $A^b$ be two adapted (w.r.t the natural filtration of $W$), cadlag, non-decreasing processes with the convention that $A^a_{0-}=A^b_{0-}=0$. Let $X=(X_t)_{t\geq 0}$ be the solution to the two-dimensional SDE: \begin{align} dX_t^a&=X_t^adW_t^a+(X^a_{t-}+X^b_{t-})dA^a_t\\ dX_t^b&=X_t^bdW_t^b+(X^a_{t-}+X^b_{t-})dA^b_t, \end{align} with initial data $X_{0-}^a=x_a>0$ and $X_{0-}^b=x_b>0$.
Since the SDE is linear, the existence and uniqueness are guaranteed. I wonder if there is any comparison theorem for such class of SDEs so that we can conclude $X^a_t>0$ and $X^b_t>0$ a.s. for all $t\geq 0$.
I have looked at Theorem 54, Philip Protter's book `Stochastic integration and Differential equations', but it only has the comparison theorem for 1-dimension and processes $A$ is assumed to be continuous.