Take the 2-minute tour ×  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.  share|improve this question It is not true
even if all $\sigma^a$ are deterministic. –  zhoraster Feb 26
'11 at 16:41  1 Answer 1  This is not true. Fix some $t>0$, and
define $t^n_k = \frac{k}{n}.t$ for $0 \leq k \leq n$. Define
$\Lambda$ as the set of deterministic, $\{-1,1\}$-valued
processes and $\Lambda_n$ as the subset of $\Lambda$ such that
$\sigma \in \Lambda_n$ if it is constant on each
$(t^n_k,t^n_{k+1})$.  Since brownian motion has unbou