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Definition (Stochastic interval)
Let η,τ\eta,\tauη,τ be stopping times. We define the stochastic interval as <spanclass="wikilink−unresolved"title="Notenotpublished">η,τ</span>={(t,ω)∈R+×Ω:η(ω)≤t≤τ(ω)}<span class="wikilink-unresolved" title="Note not published">\eta,\tau</span>=\{ (t,\omega)\in\mathbb{R}^{+}\times\Omega: \eta(\omega)\le t \le \tau(\omega) \}<spanclass="wikilink−unresolved"title="Notenotpublished">η,τ</span>={(t,ω)∈R+×Ω:η(ω)≤t≤τ(ω)}
Remark
For 0≤s<t0\le s<t0≤s<t, <spanclass="wikilink−unresolved"title="Notenotpublished">s,t</span>=[s,t]×Ω<span class="wikilink-unresolved" title="Note not published">s,t</span>=[s,t]\times\Omega<spanclass="wikilink−unresolved"title="Notenotpublished">s,t</span>=[s,t]×Ω
Stopping Time Integral