Why are we interested in whether a given hitting time is a stopping time?

Question

Why are we interested in whether a given hitting time $\tau_B=\inf\{t \geq 0\,|\, X(t) \in B\}$ of a stochastic process $\{X(t)\}_{t \in [0,\infty)}$ is a stopping time,i.e. that $\{\tau_B \leq t\} \in \mathcal{F}(t)$ holds for all $t$ (where $\{\mathcal{F}(t)\}_{t \in [0,\infty)}$ is the corresponding filtration)? If we use such a hitting time as a termination criterion (e.g. if the stock rises the first time above 200 dollars we sell) it would of course make sense to know at every time $t $ whether we have to terminate at this point. But then it would be enough to demand that $\{\tau_B=t\} \in \mathcal{F}(t)$ right?

Answer 1

Only requiring $\{\tau = t\} \in \mathcal F(t)$ for $\tau$ to be a stopping time is too weak. We usually work with augmented filtrations satisfying the usual conditions, in particular we assume that $\mathcal F(t)$ contains all null-events of $\mathcal F_\infty$. If $\tau$ is any continuous random variable with $\mathbb{P}(\{\tau = t\}) = 0$ for all $t$, then $\{\tau = t\} \in \mathcal F(t)$ for all $t$ so $\tau$ would be a stopping time under that definition. We don't want that because it would include things like the last time that $X(t)$ reaches a certain level, or the time that $X(t)$ reaches its maximum on $[0,T]$ - both of those are things that we, intuitively, shouldn't know at the time that they happen and only be able to determine later.