Why use stopping times rather than a deterministic sequence to localise a martingale?

Question

I am a beginner on stochastic processes

I am wondering why , to localise a martingale, require the existence of one non-decreasing sequence of stopping times [$ \tau_1 , \tau_2$,...] such that the corresponding stopped processes $\{M_{\tau_n \land t} \}$ are all martingales, rather than simply require the existence of one increasing infinite partitioning sequence [$t_1 \lt t_2 \lt ... \lt t_n \lt ... , t_n \rightarrow \infty$] such that the stopped processes $\{ M_{t_i \land t} \}$ are all martingales ?

I mean, what is to be gained by having stopping times rather than a deterministic sequence here ?

Answer 1

Stopping time is more general. Sometimes, you don't know what a particular time will be, e.g. the time the process hits $5$ for the first time is not constant, but is a stopping time.