Forecasts of an integrated AR(1) process (${\hat{W}}_{T,k}=W_T+k\mu +\frac{\varphi (1-{\varphi }^k)}{1-\varphi }(W_T-W_{T-1}-\mu )$)

If the increments of $W$ have mean $\mu$ and centered increments satisfy a causal autoregressive process of order one with $|\varphi |<1$, sum their recursive forecasts to obtain the displayed expression. For fixed forecast origin, ${\hat{W}}_{T,k}/k\to \mu$. One must observe the last increment, requiring $T\ge 2$ or a known initial level $W_0$; a single observed level without an initial-level model does not identify that increment.