Let $(M_t)_{t\geq0}$ be a continuous process, $\epsilon >0$ and define a sequence $(S_n)_{n\geq 0}$ by $S_{i+1}=\inf\{t > S_{i} : M_t - M_{S_i} > \epsilon \}$ and $S_0=0$.
Clearly $S_0$ is a stopping time and I would like to use induction to show the entire sequence is. I'm considering just saying that if $S_i$ is a stopping time $S_{i+1}$ is a hitting time of a open set for a continuous process - but the open set is then stochastic and I'm not sure how this affects things.
Is the argument correct? If not what can one do instead?
I'd say it was easier to do the inductive step directly.
Assume $S_i$ is a stopping time, then for every $t\in\mathbb R$ $M_{S_i\wedge t}$ is $\mathscr F_t$-measurable.
Therefore the event $$[S_{i+1}<t] = \bigcup_{0<s_1<s_2<t\in\mathbb Q} [S_i< s_1 ]\cap \left[\left|M_{S_i\wedge s_1}-M_{s_2}\right|>\varepsilon\right]\in\mathscr F_t$$