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Let $H$ be a previsible process, i.e. $H$ is measurable with respect to $\mathcal{P}$, where $\mathcal{P}=\sigma(E\times(s,t] : E \in \mathcal{F}_s; s,t \in \mathbb{R}_+)$. Need to show that $H_t$ is $\mathcal{F}_{t-}$-measurable for all $t>0$, where $\mathcal{F}_{t-}=\sigma(\mathcal{F}_s:s<t)$.

I am trying to show $\{H_t \leq r\} \in \mathcal{F}_s$, for all $s>t$, using the fact that $\{H \le r\} \in \mathcal{P}$. Now $\{H_t \le r\} = \{w | (w,t) \in \{H \le r\}\}$ and I am stuck.

Any hints?

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1 Answer 1

up vote 2 down vote accepted

These measurability questions are often easier to solve abstractly. I mean, instead of beginning with a particular predictable (previsible) process $H$ and looking into its properties; ask the question "Are there any predictable processes with the desired property"? Well, there are lots of them.

Let ${\cal K}$ consist of the set of processes formed by taking indicator functions of predictable rectangles. Every process in $K\in\cal K$ has the desired property that $K_t\in{\cal F}_{t-}$ for all $t>0$. Convince yourself that ${\cal K}$ is multiplicative, that is, $K_1,K_2\in{\cal K}$ implies $K_1K_2\in {\cal K}$. Now invoke the functional monotone class theorem, and you are almost there.

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Thank you for the link, in particular, I have never seen Theorem 2 before. However, I can only prove now that every bounded previsible process has measurable projections. What about unbounded processes? –  Tom Artiom Fiodorov Feb 4 '12 at 16:30
    
@ArtiomFiodorov Approximation by truncation. –  Byron Schmuland Feb 4 '12 at 16:33
    
brilliant! thank you –  Tom Artiom Fiodorov Feb 4 '12 at 16:53

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