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Ito's formula and proving a martingale

I have came across a similar question as q2 but I do not quite understand why $F_s(t,x,s) = 0$ for $x=s$ is enough to conclude that $F(t,B_t,S_t)$ is a continuous local martingale.

I would appreciate it if someone could explain it to me.

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    The process $S_t$ is clearly not continuous, as a Brownian motion is never monotonic on an interval . Look at the proof of Ito's lemma with jump processes, it will help / In rough words, take a subdivision $0=t_0<...$$\int_{0}^{t}{F_s(u,B_u,S_u)dS_u} \approx \sum_{k=0}^{n-1}{F_s(u,B_{t_{k}},S_{t_{k+1}})(S_{t_{k+1}}-S_{t_{k}})}$$ Take $S_{t_{k}}=max(B_{t_{0}},..,B_{t_{k-1}})$, either you have $S_{t_{k+1}}>S_{t_{k}}$ or $S_{t_{k+1}}-S_{t_{k}}$(the element in the sum is nil then). If $S_{t_{k+1}}>S_{t_{k}}$,$S_{t_{k+1}}=B_{t_{k}}$,$F_s(u,B_{t_{k}},S_{t_{k+1}})=0$ . – 2017-01-14

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