Let $X=(X_k)_{k \in \mathbb{N}}$ be a $(\mathcal{F}_k)_k$-adapted process and taking values in $\overline{\mathbb{R}}^+.$
- If $X$ is super-martingale, for every $y \in \mathbb{R}^+$ is there a uniformly bounded super-martingale $(Y^y_k)_k$ such that for all $k \in \mathbb{N},\lim_{y \to +\infty} Y_k^y=X_k$ a.s. ?
- Repeat the question considering a sub-martingale $X.$
If $X$ is a super-martingale then the super-martingale property is preserved if we consider $Y_k^y=\min(X_k,y)$ which proves 1. since for every $k \in \mathbb{N},y \in \mathbb{R},E[Y_{k+1}^y|\mathcal{F}_k] \leq \min(E[X_{k+1}|\mathcal{F}_k],y) \leq Y_k^y$ and $Y_k^y$ is uniformly bounded by $y$ and for every $k,\lim_{y \to \infty} Y_k^y=X_k.$
Any suggestions how to deal with 2. ?