I've been looking at the 2014 edition of Da Prato & Zabcyzk and the sections on predictable processes. In particular, in their Proposition 3.7 (ii), they assert that if $\Phi$ is adapted and stochastically continuous on $[0,T]$, then it has a predictable modification.

What confuses me here, is it's not clear what $\Phi$ is. In part (i), $\Phi$ is an $L(U,H)$ valued operator ($U$ and $H$ both separable Hilbert spaces). If this is the case for (ii) also, what is the topology they wish to use on the space of operators? If it is the Borel $\sigma$-algebra induced by the operator norm, won't that introduce separability issues?

If it is the strong operator topology, then I am a bit unsure of a key step in their proof. They first construct a sequence of predictable $\Phi_m$ approximations, and define the set $A$ to be points where $\{\Phi_m\}$ converge, asserting $A$ is predictable. I do not see how they get the $A$ to be in the predictable $\sigma$-algebra.

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.