I asked a question here about how to get predictions for the random-walk component of an ARIMA model.
Are there time series models in the state-space framework that might be suitable for the kind of data used with ARIMA models, perhaps for series similar to the one in the linked question? I would like to know how to build a model using state-space models rather than Box-Jenkins methods such that I can implement a Kalman filter to predict the random walk portion of the signal.
State-space models are very flexible; indeed they can encompass ARIMA models.
One class of state space models that has some overlap with ARIMA models but also has a large subset of models that don't overlap with them is the Basic Structural Model (BSM). See Harvey (1989)[1]. There are also numerous papers by Harvey (usually with other authors) relating to the BSM and at least a couple of other books. Structural models are also sometimes called unobserved components models (UCM). For example, the 1990 paper by Harvey and Peters ("Estimation Procedures for Structural Time Series Models," J. Forecasting) is not hard to locate and has some useful details that are also in the book reference I give.
Here's an outline of the Basic Structural Model:
$$y_t = \mu_t + \gamma_t +\epsilon_t,\qquad t=1,...,T$$
where $\mu_t$ is the trend component, $\gamma_t$ is a seasonal component and $\epsilon_t$ an irregular component (or noise).
The model for $\mu_t$ is:
\begin{eqnarray} \mu_t&=&\mu_{t-1}+\beta_{t-1}+\eta_t\\ \beta_t&=&\beta_{t-1}+\zeta_t \end{eqnarray}
with $\eta_t$ and $\zeta_t$ independent of each other and across time; they have mean zero and each has its own variance.
The trend component $\mu$ is "locally linear"; $\beta_t$ is the local slope.
There are several ways to write a seasonal component. The "seasonal dummy" formulation is:
$$\gamma_t=-\sum_{j=1}^{s-t}\gamma_{t-j}\,\omega_t$$
where $\omega_t$ is another independently distributed disturbance term with its own variance.
[There's also a different seasonal model that can be used based on sin and cos components.]
The parameters $\mu_t,\beta_t,\gamma_t$ form the state. The first equation is the observation equation and the remaining equations (put together) define the state equation.
See also some of the other references here
The BSM is readily extended in any number of ways, or can be made more specific by omitting unneeded components (e.g. leaving out the seasonal component if there's no seasonality), and has the nice property that its state components have nice human-understandable interpretations.
A pure random walk with noise model would set $\beta$'s and $\gamma$'s to zero:
\begin{eqnarray} y_t &=& \mu_t + \epsilon_t,\qquad t=1,...,T\\ \mu_t&=&\mu_{t-1}+\eta_t \end{eqnarray}
(and a straight-out pure random walk would set $\epsilon$ to 0).
Another paper you might find relevant is Harvey and Todd (1983) "Forecasting Economic Time Series with Structural and Box-Jenkins Models", J. Business & Economic Statistics, 1:4, since it seems to be closely related to what you are trying to do - compare state space models with ARIMA.
Many stats packages offer BSM models or something very similar; there's UCM in SAS, there's the StructTS package in R, and so on -- so you don't really have to do much to even set up the state space model (not that it's onerous).
[1]: Andrew C. Harvey (1989) Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press
