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Let ${\beta }_t$ be the cash position and ${\pi }_t$ the risky shares chosen using information at $t-1$. At the start of the period the portfolio costs ${\beta }_t+{\pi }_t{S}_{t-1}=X_{t-1}$. With cash price fixed at one, its ending value is ${\beta }_t+{\pi }_t{S}_t$. Rebalancing cannot add or remove wealth under a self-financing strategy, soThe holdings are predictable: they are chosen before the current price increment is observed. This is the discounted wealth equation in discrete time with cash as numéraire.

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).