This may be a basic question of stochastic differential equation.
Let $U$ be the closed unit disk $U=\{x \in \mathbb{R}^2 \mid |x|\le 1\}$, where $|\cdot|$ is the standard Euclidean norm on $\mathbb{R}^2$. We denote by $\partial U$ the boundary of $U$.
We consider a diffusion process $X=\{X(t)\}_{t \ge 0}$ on $U$ such that \begin{align*} X(t)=x+\int_{0}^{t} \sqrt{1-|X(s)|^2}\,dB(s),\quad t \ge 0, \end{align*} where $x \in U$ and $B=\{B(t)\}_{t \ge 0}$ is a two-dimensional Brownian motion.
My question
If $x \in \partial U$, $X(t) \in \partial U$ for any $t \ge0$ ?
If $|x|=1$, the diffusion coefficient of $X$ vanishes. Therefore, it should follows that $X(t)=x$, $t \ge 0$. However, I could not reach this equality. A simple calculation yields that \begin{align*} E[|X(t)-x|^2]&=E\left[\int_{0}^{t}(1-|X(s)|^2),ds \right]=E\left[\int_{0}^{t}\left||X(s)|^2-|x|^2 \right|\,ds \right] \\ &\le 2 E\left[\int_{0}^{t}\left|X(s)-x \right|\,ds \right],\quad t \ge 0. \end{align*} Unfortunately, this is not enough to use the Gronwall inquality. Am I overlooking something?