I want to condition a continuous-time stochastic process $X_t$ on the current value of another process $Y_t$. How can I do this?

Note: In discrete time, this is simply the conditional expectation $E(X_t | \sigma(Y_t))$ for every $t$. But in continuous time, $E(X_t | \sigma(Y_t))$ is defined only up to a null set $N_t$ which depends on $t$, and the union of $N_t$ over uncountably many $t$ can have any measure. This is why in continuous time one works with optional or predictable projections. But these are defined only when conditioning on an increasing family of $\sigma$-algebras.

I want to condition a stochastic process $X$ on the current value of another process $Y$ in a continuous-time 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?