Let $p>1$ be a real number. It is known that if $(X\_n)\_{n\geq 0}$ is a martingale bounded in $L^p$ (i.e. $\sup\{\mathbb{E}(|X\_n|^p), n\geq 0\} < +\infty$ ), then $(X\_n)\_{n\geq 0}$ converges a.s. and in $L^p$ to $X\in L^p$. This results is still true if $(X\_n)\_{n\geq 0}$ is a positive submartingale.
However, do we have such convergence results if we only suppose that $(X\_n)\_{n\geq 0}$ is a submartingale? 
Most of books only consider the case where $(X\_n)\_{n\geq 0}$ is a martingale.



Submartingales bounded in $L^p$, $p>1$
The almost sure convergence is true for any submartingale with an integrable upper majorant. This implies the $L^1$ convergence through uniform integrability. 


However, in general, there is no convergence in $L^p$. I'll give a counterexample for $p=2$, but it can be modified for any $p>1$. 


Let $X\_n = - e^{W\_n - n}$, where $W$ is the standard Wiener process. This is a submartingale with $E[X^2\_n] = 1$, and $X\_n\to 0$, $n\to\infty$, almost surely. But there is no convergence in $L^2$.


