Definition ($L^{p}$-martingale)

Let $(X_{t})_{t\ge 0}$ be a $(\mathcal{F}_{t})_{t\ge 0}$-adapted process. $(X_{t})_{t\ge 0}$ is called an $L^{p} \ \ (\mathcal{F}_{t})_{t\ge 0}$-martingale if

1. $X_{t}\in\mathscr{L}^{p}(\Omega,\mathcal{F},P), \ \forall t\ge 0$ (this implies $X_{t}\in\mathscr{L}^{1}$) and;
2. $\forall_{0}\le s\le t: X_{s}=E[X_{t}|\mathcal{F}_{s}]$ a.s.