Is there any way to decide whether for $X_n$, $Y_n$ martingales for the same filtration,
$$Z_n = (X_n+Y_n)1_{Y_n>0} + X_n1_{Y_n\leq0}$$
is a martingale, sub-martingale or super-martingale?
Obviously, $Z_n$ has finite expectation and is measurable w.r.t the given filtration, however, I cannot seem to find a way to proof any of the martingale properties.
We may write $Z_n = X_n + \max(0,Y_n)$, and since both the sum and the maximum of two submartingales is a submartingale, $Z_n$ is a submartingale.
On the other hand, if we set $X_n=0$ and let $Y_n$ be a symmetric, simple random walk starting at $Y_0=0\in \mathbb{Z}$, then we can check directly that $E[Z_1] > 0 = E[Z_0]$. Therefore $Z_n$ is neither a supermartingale nor a martingale.