Relating Discrete ARMA Models to Continuous-Time Processes
Summary
The document explores how discrete ARMA and seasonal autoregressive models might relate to continuous-time processes. One proposal replaces the discrete lag coefficients with continuous lag-weighting functions and represents the lagged contributions through integrals. Interpolation is suggested for constructing these functions from discrete coefficients, though the answer raises uncertainty about how discrete innovations should scale with Brownian motion and how to express the resulting stochastic differential equation.
A more concrete special case connects an AR(1) model to an Ornstein–Uhlenbeck process by scaling its drift and noise terms with the time step. A seasonal component is then sketched as a periodic process. These are conceptual correspondences, not a general conversion method: the discussion leaves higher-order autoregressive models and seasonal moving-average dynamics unresolved, and flags uncertainty in the proposed continuous-time formulation.
Key ideas
- A proposed ARMA extension replaces discrete lag coefficients with continuous weighting functions over finite lookback periods.
- Interpolation can map discrete autoregressive and moving-average coefficients into continuous lag functions.
- An AR(1) process can approximate an Ornstein–Uhlenbeck process when its drift and noise are scaled with the time step.
- The discussion does not establish a general continuous-time equivalent for higher-order or seasonal ARMA models.
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# From discrete time series models to continuous
# From discrete time series models to continuous
Is it possible to convert an SARIMA model to a continuous model?
If so, what is the methodology to do that?
## Answer by user25064 (score 1, accepted)
https://quant.stackexchange.com/a/27443
Let's look at the formula for an ARMA(p, q) model
$$ X_t = c + \sigma z_t + \sum_{i=1}^p \varphi_i X_{t-i} + \sigma\sum_{i=1}^{q}\theta_i z_{t-i} $$
where $z_t \in \mathcal{N}(0,1)$ for all $t$. Transform this into the continuous time counterpart below.
$$ X_t = c + \sigma W_t + \int_0^{T_p}\varphi(v)X_{t-v} dv + \sigma\int_0^{T_q}\theta(v)W_{t-v}dv $$
Where $T_{p,q}$ are the width of the lookback window (in years) for the AR and MA terms. The sequence of coefficients have been replaced by continuous functions of time $\varphi, \theta : \mathbb{R_+} \mapsto \mathbb{R}$. Interpolation methods can be used to go from the discrete $\varphi_i$ to the continuous $\varphi(v)$.
A problem that I see with this is that I believe the $z_t$ should probably not be replaced with $W_t$ but something more like "$\sqrt{dt}dW_t$ " but I am not sure how to formalize that. I am also not sure about how to write out the total derivative for this SDE $dX_t$ any help with that would be appreciated.
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/26418
You find the connection between an AR(1) and an Ornstein-Uhlenbeck process here in QSE if you search.
Taken the description form here (which is just the way to do it):
$$ X_{n+1} = c + a X_n + b \varepsilon_k $$ and by setting $c=\theta \mu \Delta t, a=−\theta \Delta t$ and $b =\sigma \sqrt{\Delta t}$ you will get the discrete time approximation $$ X_{n+1} = \theta(\mu - X_n)\Delta t + \sigma \varepsilon_k\sqrt{\Delta t}, $$ which corresponds to the continuous time OU-process.
If you add a seasonal process then you have SAR(1).
Thus AR(1) + seasonal component in discrete time is $$ dX_t = a (b-\mu) dt + dB_t + dS_t $$ where $S_t$ is a periodic function. This would be the solution for the AR(1) case and no seasonal AR or MA component.
If we think of the dependence structure in higher order AR processes I wonder whether there is a continuous time model for AR(n) for $n>1$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.