 Solution

Condition (i) says that each discounted risky price $X^i$ is an integrable $\mathbb{Q}$-martingale. For a bounded predictable risky holding process $\theta$, self-financing gives the discounted wealth equation in discrete timeEach increment is integrable andHence $V$ is a $\mathbb{Q}$-martingale, proving (i) $\Rightarrow$ (ii).