I'm having difficulty finding papers which deal with the following inversion problem.
Suppose I have a stochastic process $Y_t$ (which is described by a certain Hilbert-Space-valued SDE). I want to know how to characterize all stochastic processes $X_t$ satisfying the following:

  • If $\mathfrak{G}_t$ is the filtration generated by $Y_t$ and $\mathfrak{F}_t$ is the filtration generated by $X_t$, then $$ \mathfrak{G}_t\subseteq \mathfrak{F}_t$$
  • $\mathbb{E}[X_t|\mathfrak{F}_t]=Y_t$.

I expect this has something to do with inverting the conditional expectation given $\mathfrak{F}_t$, but how can I do that?

  • 1
    $\begingroup$ something is funny about this because $X_t$ is $\mathfrak{F}_t $ measurable. $\endgroup$ – user83457 Nov 28 '16 at 8:06

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.