 Solution

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.