= Solution
Use a <telescoping replication of a stock-price sum>. Hold $T-t+1$ shares during interval $(t-1,t]$; at time $t$, sell one share and keep its proceeds in the bond. Start with $T$ shares and no cash, costing $TS_0$.
After the time-$t$ rebalance, the <stock> holdings are $T-t$ and the cash holdings are $\sum_{u=1}^tS_u$, so wealth is
$$
V_t=\sum_{u=1}^tS_u+(T-t)S_t.
$$
The sale of one share exactly funds the cash increase, making the strategy <self-financing>. Equivalently,
$$
\boxed{TS_0+\sum_{t=1}^T(T-t+1)(S_t-S_{t-1})
=\sum_{u=1}^TS_u.}
$$
At time $T$ there are no remaining shares, and the cash equals the claim. All <stock> positions over trading intervals are predictable.
Back to article page
