https://i.stack.imgur.com/gvLDp.png
Given the probability space in the attached image I would like to Show that τ is a stopping time with respect to FX.
I have calculated the FX to be:
F0X=σ({X0})=(ØΩ)
F1X= σ({X0,X1)}= σ({a,b},{c,d})
F2X= σ(X0,X1,X2})= σ({a},{b},{c},{d})=2Ω
and I know from here my next step would be to calculate τ of each event. This is where I come unstuck as it appears that τ is actually 0,0,0,0 because at Time 0 the value is already above the stopping time of 10 therefore we never reach event 1 or 2. But if τ = 0 my understanding is it does not fit the criteria to be a stopping time. Excuse the beginner level question but can anyone help me work through this problem?
Many thanks