I want to condition a stochastic process $X$ on the current value of another process $Y$ in a continuoustime setting. So I am looking for a process $Z$ such that for every fixed $t$, $$Z_t = E\big(X_t \big \sigma(Y_t)\big) \quad \text{a.s.}\tag{1}$$ Uniqueness holds when $Z$ is required to be cadlag. Alternatively, uniqueness follows when $Z$ is required to be optional and $$Z_{\tau} 1_{\{\tau<\infty\}}= E\big(X_{\tau}1_{\{\tau<\infty\}} \big \sigma(Y_t)\big) \quad \text{a.s.}\tag{2}$$ holds for every stopping time $\tau$. Existence follows by a monotone class argument for $X$ if I can prove that the process $$X(t,\omega)=1_{[a,b)}(t) 1_F(\omega)$$ admits $Z$ as in $(1)$ or $(2)$. Here $[a,b)$ is some interval and $F$ a measurable set. But I am not able to show this. Any hints or ideas?
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