 Solution

Difference the local-level state-space model to eliminate its latent level:Independence and Gaussianity give a stationary Gaussian process withFor $V,W>0$, letThese satisfy $q(1+{\vartheta }^2)=W+2V$ and $q\vartheta =-V$. Therefore $D$ has the covariance of the moving-average model ${\epsilon }_t+\vartheta {\epsilon }_{t-1}$, where $\epsilon$ is Gaussian white noise of variance $q$. This is an actual representation, not just covariance matching: define ${\epsilon }_t={\sum }_{j\ge 0}(-\vartheta {)}^j{D}_{t-j}$. The convergent inverse filter has constant spectral density $q/(2\pi )$, so its Gaussian coordinates are independent, and multiplication by $1+\vartheta B$ recovers $D$.

ThusThis is the formal ARMA equation with autoregressive coefficient one. Its unit root means that the levels are generally not stationary: if the initial state has finite variance and is independent of future noises, $\mathrm{Var}(S_t)=\mathrm{Var}(S_0)+tW$. In stationary-model terminology the correct classification is ARIMA(0,1,1). The requested ARMA description must therefore allow a unit-root equation.

When $V=0$, the differences are just $w_t$; when $W=0$ and $V>0$, they are $v_t-v_{t-1}$, a noninvertible MA(1) with coefficient $-1$. If both variances vanish, the model is deterministic apart from a possible initial random level. These limits explain the qualifications on the invertible representation above.