Is there an example of an $L^p$-bounded martingale (discrete-time/continuous-time) that is not $L^{p+\epsilon}$-bounded for any $\epsilon>0$?
For discrete time, my idea is to use $M_n = \prod_{j=1}^n \xi_j$ where $\xi_j$ are iid non-negative random variables with $\mathbb{E}\xi_j = 1,\mathbb{E}\xi_j^p = 1, \mathbb{E} \xi_j^{p+\epsilon}>1$.
What about continuous-time?