19.2 Random walk

A non stationary stochastic process $(Y_t)$ is a random walk when:

$$Y_t=Y_{t-1} + \omega_t$$

• A random walk is a stochastic process whose first difference is a white noise, i.e.

$$Y_t - Y_{t-1}=\omega_t$$ * Sometimes an additional parameter is included:

$$Y_t = \mu + Y_{t-1} + \omega_t$$

• A random walk can be written as:

$$\nabla Y_t = \mu + \omega_t,$$

so that a non-stationary stochastic process $(Y_t)$ is a random walk when its regular difference of order 1 $\left(Y_{t} \right ) \equiv \left ( Y_{t}-Y_{t-1} \right )$ is a stationary process.