$L^p$ Boundedness of a martingale

Question

So I recently read a paper where the authors claim that if for some martingale $(M_t)_{t\geq 0}$ we have $$\mathbb E[M_{t+s}^p]-\mathbb E[M_s^p]\leq \exp(-cs) (\mathbb E[M_{t}^p ]-\mathbb E[M_t]^p)$$ for some $p\in (1,2], c>0$ and any $s,t\geq 0$ then it automatically follows that the martingale is bounded in $L^p$. In some other paper it was elaborated that this implies that $$\sup_{t \geq 0}\sum_{n=0}^\infty \mathbb E[M_{t+(n+1)s}^p]-\mathbb E[M_{t+ns}^p]< \infty.$$ However, for neither of those inequalities it seems obvious to me that they imply $L^p$ boudedness. Does anyone have any ideas why this is true?

Answer 1

I assume the martingale is positive, other the powers do not make sense. In particular, $\mathbb{E}(M_t)^p = \mathbb{E}(M_0)^p \geq 0$, and the inequality given implies that for any $s,t \geq 0$, $$ \mathbb{E}(M_{t+s}^p) - \mathbb{E}(M_s^p) \leq e^{-cs} \mathbb{E}(M_t^p). $$ If you take $t=1$, $s=k$, then this gives $$ \mathbb{E}(M_{k+1}^p) - \mathbb{E}(M_k^p) \leq e^{-ck} \mathbb{E}(M_1^p), $$ so after summing from $0$ to $n$ $$ \mathbb{E}(M_{n+1}^p) \leq \mathbb{E}(M_0^p) + \frac{1-e^{-c(n+1)}}{1-e^{-c}} \mathbb{E}(M_1^p) \leq \mathbb{E}(M_0^p) + \frac{1}{1-e^{-c}} \mathbb{E}(M_1^p). $$ This gives that it is bounded on integers, which is enough since $(M_t^p)$ is a supermartingale and thus $t \mapsto \mathbb{E}(M_t^p)$ is increasing.