ARIMA Model subsumes ARMA Model

Theorem
Let $S$ be a stochastic process based on an equispaced time series.

Let $M$ be an ARMA model for $S$.

Then $M$ is also an implementation of an ARIMA model.

Proof
Let the values of $S$ at timestamps $t, t - 1, t - 2, \dotsc$ be $z_t, z_{t - 1}, z_{t - 2}, \dotsc$

Let $\tilde z_t, \tilde z_{t - 1}, \tilde z_{t - 2}, \dotsc$ be deviations from a constant mean level $\mu$:
 * $\tilde z_t = z_t - \mu$

Let $a_t, a_{t - 1}, a_{t - 2}, \dotsc$ be a sequence of independent shocks at timestamps $t, t - 1, t - 2, \dotsc$

By definition of ARMA model, $M$ is implemented as:


 * $(1): \quad \tilde z_t = \phi_1 \tilde z_{t - 1} + \phi_2 \tilde z_{t - 2} + \dotsb + \phi_p \tilde z_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$

The general ARIMA model is implemented as:


 * $w_t = \phi_1 w_{t - 1} + \phi_2 w_{t - 2} + \dotsb + \phi_p w_{t - p} + a_t - \theta_1 a_{t - 1} - \theta_2 a_{t - 2} - \dotsb - \theta_q a_{t - q}$

where:
 * $w_t = \nabla^d z_t$

Letting $d = 0$ we have:
 * $w_t = z_t$

Setting $w_t = z_t - \mu = \tilde z$, we recover $(1)$.