 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 $T{S}_0$.

After the time-$t$ rebalance, the stock holdings are $T-t$ and the cash holdings are ${\sum }_{u=1}^t{S}_u$, so wealth isThe sale of one share exactly funds the cash increase, making the strategy self-financing. Equivalently,At time $T$ there are no remaining shares, and the cash equals the claim. All stock positions over trading intervals are predictable.