 Locally defined stochastic process

A continuous locally defined process is a pair $(X,T)$ specified on the stochastic interval $[0,T)$, with a lifetime stopping time $T$ approached by an announcing sequence for a stopping time $T_n$. Each stopped process is an ordinary continuous adapted process. This is the convention used for continuous local differential equations; no value at $T$ is required. Restricting a globally defined process gives examples, but finite lifetimes also allow approach to a domain boundary or explosion.