Let suppose there is a sequence of random variables $(X_n)_n$, with some $X_0 \gg 1$ being a constant and $\mathbb{E}(X_n^2 | X_{n-1}) = f(X_{n-1})$, where $f(X_{n-1}) < X_{n-1}^2$, while $X_{n-1} > 2$. We know $\mathbb{E}(X_n) < \infty$ and even $X_n < X_0 + n$ a.s.
For example, $\mathbb{E}(X_n^2 | X_{n-1}) = X_{n-1}^2 - X_{n-1}$.
This somehow resembles supermartingale, but is it?
From $\mathbb{E}(X_n^2 | X_{n-1}) = f(X_{n-1})$
What is the rigorous way to get to $\mathbb{E}(X_n | X_{n-1})$ ?
What I mean, is there any general procedure/algorithm, which given $\mathbb{E}(X_n^2 | X_{n-1}) = f(X_{n-1})$ produces $g(X_{n-1})$, s.t. $\mathbb{E}(X_n | X_{n-1}) = g(X_{n-1})$.
Let suppose this is possible and $(X_n)_n$ is supermartingale.
Let $\tau = \inf \{n: X_n < 2\}$.
How can I find $\mathbb{E}(\tau)$? i.e. is there any useful "Optional Stopping Theorem", or are "Wald's-like identities" extremely problem specific?