Linking the orders of an ARMA and ARIMA representation of a unit root process

If $Delta Y_t$ can be described by a (stationary) $ARMA(p,q)$ process, then $Y_t$ can be described by an $ARIMA(p,1,q)$ process. Theoretically (concerns of usefulness aside), does this mean we can consider $Y_t$ as a (non-stationary) $ARMA(p+1,q)$ with a unit root in its autoregressive component?
I’m trying to get a sense of how autoregressive, moving average, and integrated orders of a unit root process are linked to each other in its $ARMA$ and $ARIMA$ representations. To my understanding, it’s purely an issue of notation. If it’s not, please let me know.