My question is as follow:
Let $(\Omega,\cal{F}_\infty,\{\cal{F}_t\},\mathbb{P})$ be the filtred probability space. Further, denote $\cal{F}^*_t$ as the usual augmented filtration. Now, given a continuous $\cal{F}^*$-progressively measurable process $X$, we define: $H:=\inf\{t|X_t\leq 0\}$. Since $X$ is continuous, $H$ is an $\cal{F}^*$-stopping time. My question is whether there exists an $\cal F$-stopping time $\hat H$ such that $H=\hat H$ a.s. ? Thank you !