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.