Consider a nonlinear stochastic differential equation of Ito type
$$ dx = f(t,x)dt + g(t,x)dB, \qquad (1) $$
where B is a p-dimensional Browninan motion, and $x\in\mathbb{R}^n$. In the literature I have found there is always assumed that $x\in\mathbb{R}^n$. Then, theorems for the existence and uniqueness of solutions of (1) are derived. Also Lyapunov stability theorems are extended to the stochastic case.
Then I wonder if the results of existence, uniqueness of solutions, and stochastic stability of (1) still apply when $x$ belongs to a certain subset of $\mathbb{R}^n$, i.e. $x\in\Phi\subseteq\mathbb{R}^n$. Is it possible to use the definition of forward invariant spaces as in the deterministic case?