Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge 0}$ be a filtration on $(\Omega,\mathcal A)$
- $(E,\left\|\;\cdot\;\right\|)$ be a separable Banach space
- $(X_t)_{t\ge 0}$ be an $\mathcal F$-adapted and integrable real-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)$
- $(Y_t)_{t\ge 0}$ be an $\mathcal F$-adapted and Bochner integrable $E$-valued stochastic process on $(\Omega,\mathcal A,\operatorname P)$
If $X$ is a $\mathcal F$-martingale and $\tau$ is a $\mathcal F$-stopping time, then $X^\tau$ is a $\mathcal F$-martingale by the optional stopping theorem.
We can show that:
Let $E^\ast$ be the dual space of $E$. Then, $Y$ is a $\mathcal F$-martingale $\Leftrightarrow$ $$f(X)\text{ is a }\mathcal F\text{-martingale}\;\;\;\text{for all }f\in E^\ast\tag 1\;.$$
How can we use this statement, in order to show, that the optional stopping theorem applies to $E$-valued processes like $Y$, too?