Let $(Y^a: a\in \Lambda)$ be a set of random processes given by $$Y^a(s) = \int_0^s \sigma^a(r) dW(r)$$ where $W$ is Brownian motion w.r.t. filtered probability space $(\Omega, \mathcal{F}, P, \mathcal{F}_t)$, $\sigma^a(\cdot)$ is uniformly bounded predictable process, s.t. $|\sigma^a(r)|<1$ for all $r$ and $a$. Let $\theta^a = \inf(s>0, Y^a(s) \ge 1)$ be a stopping time.

[Q.] Is the following true, $$\inf_{a\in \Lambda} (\theta^a) >0, \quad a.s.-P$$

When $\Lambda$ is finite set, it is clearly true. But, I am not sure otherwise. Thanks.