Let $b: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline b,\overline b]$ and $a: \mathbb R_+\times\mathbb R\times [0,1]\to [\underline a,\overline a]$ be Lipschitz, where $\overline b>\underline b>0$ and $\overline a>\underline a>0$ are fixed constants. Denote by $X^m$ the (unique strong) solution of the SDE
$$X^m_t=1+\int_0^t b(s,X^m_s,m)ds + \int_0^t a(s,X^m_s,m)dW_s,\quad \forall t\ge 0,$$
where $m\in [0,1]$ is any constant. Set $\tau_m:=\inf\{t>0: X^m_t\le 0\}$. It follows from Gronwall's inequality that, there exists $C>0$ s.t.
$$\mathbb E[\sup_{0\le s\le t}|X^m_s-X^n_s|^2] \le Cte^{Ct}|m-n|^2,\quad \forall m,n\in [0,1].$$
Can we show the existence of $\alpha>0$ and $C_{\alpha}>0$ s.t.
\begin{eqnarray*} \mathbb E[|\tau_m\wedge t -\tau_n\wedge t|] &\le& C_{\alpha}|m-n|^{\alpha} \\ \big|\mathbb P[\tau_m>t] -\mathbb P[\tau_n>t]\big| &\le& C_{\alpha}|m-n|^{\alpha}? \end{eqnarray*}
Here $x\wedge y:=\min(x,y)$. Any answer, comments or references are appreciated.
