question about martingale inequality (similar to the condition of submartingale convergence)

Question

if $X_n$ is a submartingale which can be decomposed as $X_n=M_n-N_n$, where $M_n$ is a martingale, and $N_n$ is a non-negative supermartingale.

Can we get the conclusion that $\sup_n\mathbb E X_n^+<\infty$?

I have tried to show that $\sup_n\mathbb E|X_n|<\infty$ since we can get $\sup_n\mathbb E X_n<\infty$ easily. But it seems no reason to be true. So is there a counterexample? thanks.

Answer 1

Taking $X_n$ to be simple random walk is a counterexample. Since $X_n$ is a martingale we have the decomposition with $N_n = 0$. It's known that $E|X_n| \approx \sqrt{\frac{2n}{\pi}}$ (see MathWorld, eq. (39)), and by symmetry $EX_n^+ = \frac{1}{2} E|X_n|$ so $\sup_n E X_n^+ = \infty$.