I am reading a proof of reflection principle of Wiener process, and it seems that the proof assumes a notation which appeared earlier in the note but not specify in the statement of the theorem.
The note states that:
With $\{W_{t}, t\geq 0\}$ the standard Wiener process, let $M_{t}:=\sup_{s\in[0,t]}W_{s}$ denote its running maxima and $T_{b}:=\inf\{t\geq 0:W_{t}=b\}$ the corresponding passage times. Then for any $t,b>0$, we have $$\mathbb{P}(M_{t}\geq > b)=\mathbb{P}(\tau_{b}\leq t)=\mathbb{P}(T_{b}\leq t)=2\mathbb{P}(W_{t}\geq b).$$
One could see that the author does not specify what is $\tau_{b}$. Then he argued as follows:
By (some proposition), $\tau_{b}$ is a stopping time for $\mathcal{F}_{t}^{W}$. Further, since $b>0=W_{0}$ and $s\mapsto W_{s}$ is continuous, clearly $\tau_{b}=T_{b}$.
In the proposition he refers to, the first hitting time $\tau_{B}:=\inf\{t\geq 0:X_{t}\in B\}$ appears. Thus, I guess here he means $\tau_{b}=\tau_{\{b\}}$.
But do really need such an argument to show $\tau_{\{b\}}=T_{b}$? Isn't $W_{t}\in\{b\}$ equivalent to $W_{t}=b$? Do we have exception when the trajectory is not continuous?
Is there other possible and plausible guess for $\tau_{b}$?
Thank you!
Of course, $W_t \in \{b\}$ is same as $W_t=b$ whether or not $W_t$ is continuous. My guess is $\tau_b=\inf \{t: W_t \geq b\}$. With this definition you need continuity to say that $\tau_b=T_b$.