= 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 $\beta_t+\pi_tS_{t-1}=X_{t-1}$. With cash price fixed at one, its ending value is $\beta_t+\pi_tS_t$. Rebalancing cannot add or remove wealth under a <self-financing strategy>, so
$$
\boxed{X_t=X_{t-1}+\pi_t(S_t-S_{t-1}).}
$$
The 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>.
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