# 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?