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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?

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

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