= Forecasts of an integrated AR(1) process
{title2=$\widehat W_{T,k}=W_T+k\mu+\frac{\phi(1-\phi^k)}{1-\phi}(W_T-W_{T-1}-\mu)$}
If the increments of $W$ have mean $\mu$ and centered increments satisfy a causal <autoregressive process> of order one with $|\phi|<1$, sum their recursive forecasts to obtain the displayed expression. For fixed forecast origin, $\widehat W_{T,k}/k\to\mu$. One must observe the last increment, requiring $T\ge2$ or a known initial level $W_0$; a single observed level without an initial-level model does not identify that increment.
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