Theorem (Martingale Equivalence for Stopping Times)

Let $(X_{t})_{t\ge 0}$ be right continuous and $(\mathcal{F}_{t})_{t\ge 0}$-adapted. Then $(X_{t})_{t\ge 0}\text{ is }(\mathcal{F}_{t})_{t\ge 0}$-martingale if and only if $\forall$ bounded $(\mathcal{F}_{t})_{t\ge 0}$-stopping times $T$:

1. $X_{T}\in\mathscr{L}^{1}(\Omega,\mathcal{F},P)$
2. $E[X_{T}]=E[X_{0}]$