Telescoping replication of a stock-price sum ($T{S}_0+{\sum }_{t=1}^T(T-t+1)\Delta S_t={\sum }_{t=1}^T{S}_t$)

With a constant cash account, holding $T-t+1$ shares during interval $(t-1,t]$ and selling one share at each endpoint replicates ${\sum }_{t=1}^T{S}_t$. Each sale is retained in cash; the initial cost is $T{S}_0$. The proof is a pathwise telescoping identity.