Stochastic differential equations depending on a parameter

We consider a stochastic differential equation
$$ d\xi_\theta=a_\theta(t,\xi_\theta( \cdot )) dt+B_\theta(t,\xi_\theta(t)) dw(t),\qquad\xi_\theta(0)=x_\theta, $$
such that its coefficients and initial condition are continuous functions of $\theta\in\Theta$, where $\Theta$ is a complete metric space. If an equation has a strong solution on a dense subset $\Theta_1\subset\Theta$, then $\Theta_1$ is of the second category and coincides with the set $\Theta_0$ of continuity of $\xi_\theta(t)$.