I know the stochastic filtering problem estimates the dynamics of the density $\pi_t(\phi)$ of the random variable $$ \mathbb{E}\left[ \phi(X_t) \mid \mathfrak{F}_t^Y \right]^o, $$ where $X_t$ is the signal process, $\mathfrak{F}_t^Y$ is the filtration generated by the observation process $Y_t$ at time $t$, $\phi$ is a suitable $C^{2}$ function and $^o$ is the optional projection.
My question is, what is stochastic smoothing? I read that it has something to do with using "future" information, but in what sense, rigorously speaking? Do we look at another projection, besides optional one, or do we instead condition on the $\sigma$-algebra $\mathfrak{F}_{\infty}^Y$?
The smoothing problem is naturally related to the filtering problem; it is the problem of finding the conditional expectations of the form $$ \mathbb{E}\left[ \phi(X_t) \mid \mathfrak{F}_T^Y \right],$$ where $0\leq t\leq T$ and $\phi$ is a measurable function such that $\phi(X_t)$ is an integrable random variable. The optional projection is not required if $T$ is fixed. It is my understanding that the optional projection in filtering is required in order that the filter estimate does not anticipate any jumps that might occur because of jumps in the observation process. For a filter that runs in real time, this is very important. Under some conditions, this can be reduced to the problem of finding $$\mathbb{E}\left[ \phi(X_t) \mid \mathfrak{F}_{t^-}^Y \right].$$
In constrast, smoothing usually occurs as an offline process that corresponds to the scenario of receiving all the data that has been collected by observing the process $Y$ over the interval $[0,T]$, and using it to estimate the hidden state $X_t$ at some time in $t\in[0,T]$.