fa.functional analysis – Solution of SDE system with each equation having strong unique solution

Suppose that we have a system of SDEs

$$
d X_t = a(X_t, Y_t)dt + b(X_t, Y_t) dB^x_t,\
d Y_t = c(Y_t)dt + d(Y_t) dB^y_t,
$$

Also let

  1. The initial condition $X_0$, $Y_0$ and filtrations generated by $B^x_t$, $B^y_t$ are mutually independent. $B^x_t$ and $B^y_t$ are also independent.
  2. $Y_t$ has strong unique solution with respect to $Y_0$ and $B^y_t$.
  3. For every realization of $Y_t$ (i.e., fix $Y_t$), the solution of $X_t$ is strong and unique.

Question is, is the solution to the entire SDE system is also strong and unique? If not what other conditions are missing?

I think the underlying solution probability space is the Cartesian of the solution probability spaces of these two sub-SDEs, should it be?