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pr.chance – Predictability of countably valued accessible stopping occasions on full and cadlag filtrations Answer

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pr.chance – Predictability of countably valued accessible stopping occasions on full and cadlag filtrations

The following query is motivated by this sever of the proof of Lemma 2 on web page 107 of the bespeak Stochastic integration and differential equations of Philip Protter.

Lemma 2. Let $T$ breathe a very inaccessible stopping time. For $delta > 0$, let $R(delta) = sup_{t leq v} P(t leq T leq t + delta vert mathcal{F}_{t})$. Then $R(delta) to 0 $ in chance as $delta to 0$.

Proof of Lemma 2. Let $a >0$ and $S_{n}(delta) = inf lbrace t in D_{n}: P(t leq T leq t + delta vert mathcal{F}_{t}) > a rbrace wedge v.$ First we occupy that $S_{n}(delta)$ is lower than $T$. Since $S_{n}$ is countably valued, it’s accessible, and since $T$ is completely inaccessible, $P(S_{n}(delta) = T)=0$. Suppose that $Gamma subset lbrace T< t rbrace,$ and too $Gamma in mathcal{F}_{t}$. Then

commence{align}
Eleft[ Eleft[ 1_{lbrace t leq T leq t+ delta rbrace}vert mathcal{F}_{t}right] 1_{Gamma}privilege] = E left[ 1_{lbrace t leq T leq t+ delta rbrace} 1_{Gamma} right] = 0
aim{align}

$cdots$

Is each countably accessible stopping time (in a whole and cadlag filtration) a predictable stopping time?

I question this as a result of what I need to do is to show that $S_{n}(δ)$ is predictable (I suppose utilizing the truth that it’s accessible and countably valued), and subsequently I can employ the speculation that T is completely inaccessible to show $P(S_{n}(δ)=T)=0$

Definition. A stopping time $T$ is completely inaccessible if for each predictable stopping time $S$,

commence{align}
Plbrace w: T(w) = S(w) < infty rbrace = 0
aim{align}

Any reference or intimate will breathe welcome.

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