Let $\sigma:\mathbb R_+\times\mathbb R\to [1,2]$ be measurable. Consider the SDE $dX_t = \sigma(t,X_t)dW_t$, where $X_0>0$ is independent of Brownian motion $(W_t)_{t\ge 0}$. For every $T>0$ and $R>0$, can we always show

$$\mathbb P[\inf_{0\le t\le T}X_t\le 0]>0 \mbox{ and } \mathbb P[X_T>R]>0?$$

Here we assume the existence of the solution to the above SDE.

Let $\sigma:\mathbb R_+\times\mathbb R\to [1,2]$ be measurable. Consider the SDE $dX_t = \sigma(t,X_t)dW_t$, where $X_0>0$ is independent of Brownian motion $(W_t)_{t\ge 0}$. For every $T>0$ and $R>0$, can we always show

$$\mathbb P[\inf_{0\le t\le T}X_t>0]>0 \mbox{ and } \mathbb P[X_T>R]>0?$$

Here we assume the existence of the solution to the above SDE.