Let $\mathcal X=(X_i)_{i\in I}$ be a family of integrable, real valued random variables on $(\Omega,\mathcal F,\mathbb P)$. One easily sees that $\mathcal X$ is uniformly integrable iff both $\mathcal X^+=((X_i)^+)_{i\in I}$ and $\mathcal X^-=((X_i)^-)_{i\in I}$ are uniformly integrable.
Question :
Suppose $\mathcal S=(S_t)_{t\geq 0}$ is a submartingale. Is $\mathcal S^-$ automatically uniformly integrable ? i.e. is $\mathcal S$ uniformly integrable iff $\mathcal S^+$ is uniformly integrable ?
Note :
- there is a filtration lurking in the background which I did not mention, and didn't a priori impose any assumptions on. Feel free to do so if needed.
- I am interested both in the discrete and continuous time case, with $\mathcal S$ (right) continuous if needed.
- The question is inspired by a recent question of mine about the uniform integrability of the family of stopped processes of a uniformly integrable submartinale. I didn't write down the details, but it seems that, if true, the above may facilitate the application of, say, the de la Vallée-Poussin criterion for uniform integrability.
- Part of the motivation for this question stems from the observation that a submartinale $\mathcal S$ is bounded in $L^1$ iff $\mathcal S^+$ is bounded in $L^1$. Of course being bounded in $L^1$ is necessary for being uniformly integrable.