Expected value of a random walk with drift

Question

A random walk with drift is Xt = 0.2 + X{t-1} + \varepsilont with X0 = 0 and white-noise shocks. Find E[X5].

Accepted Answer

Each step adds a fixed drift of 0.2 plus a mean-zero shock. Unrolling the recursion: Xt = 0.2\,t + (\varepsilon1 + \dots + \varepsilont). Taking expectations, the shocks average to zero and only the drift survives: E[Xt] = 0.2\,t. At t = 5: E[X5] = 0.2 	imes 5 = 1. The mean grows linearly with time, so a random walk with drift is nonstationary in its mean (on top of the variance still growing as t\sigma^2). To make it stationary you difference it: Xt - X{t-1} = 0.2 + \varepsilont, which is white noise around the constant 0.2. Takeaway. Drift turns the accumulating shocks into a rising trend:…

Expected value of a random walk with drift

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