This question can be seen as a continuation of Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$

Consider the martingale given as

$$X_t=1+\int_0^t a(s,X_s)dW_s,\quad \forall t\ge 0.$$

Denote $\tau:=\inf\{t\ge 0: X_t\le0\}$. My question is whether $t\mapsto \mathbb P[\tau>t]$ is Holder continuous, i.e. $\exists C>0, \alpha\in (0,1]$ s.t.

$$0\le \mathbb P[\tau>t]-\mathbb P[\tau>t+\Delta t]\le C\Delta t^{\alpha}.$$

Any solution, references or comments are appreciated. Here we assume that $0<\underline a \le \inf_{(t,x)} a(t,x)\le \sup_{(t,x)} a(t,x)\le \overline a$, $t\mapsto a(t,x)$ is continuous and $|a(t,x)-a(t,y)|\le L|x-y|$ for some $L>0$.

PS : My idea is as follows :

\begin{eqnarray} \mathbb P[\tau>t]-\mathbb P[\tau>t+\Delta t] &=& \mathbb P\left[\inf_{0\le s\le t}X_s>0, X_t+\inf_{t\le u\le t+\Delta t}\int_t^ua(s,X_s)dW_s\le 0\right] \\ &=& \mathbb P\left[\inf_{0\le s\le t}X_s>0, X_t>0, X_t+\inf_{t\le u\le t+\Delta t}\int_t^ua(s,X_s)dW_s\le 0\right] \\ &\le &\mathbb P\left[X_t>0, X_t+\inf_{t\le u\le t+\Delta t}\int_t^ua(s,X_s)dW_s\le 0\right] \\ &=& \int_{(0,\infty)}\mathbb P\left[x+\inf_{t\le u\le t+\Delta t}\int_t^ua(s,X_s)dW_s\le 0\Big|X_t=x\right]\mathbb P[X_t\in dx]. \end{eqnarray}

Therefore, it suffices to estimate the conditional probability

$$\mathbb P\left[x+\inf_{t\le u\le t+\Delta t}\int_t^ua(s,X_s)dW_s\le 0\Big|X_t=x\right]$$