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The continuous-time (linear) state space model can be written

\begin{align*} \text{d}\mathbf{x}_t &= \mathbf{F} \,\mathbf{x}_t \, \text{d}t + \mathbf{G} \,\text{d} \boldsymbol{\beta}_t \\ \text{d} \mathbf{z}_t &= \mathbf{H} \,\mathbf{x}_t \, \text{d}t + \text{d}\boldsymbol{\eta}_t \end{align*}

where $\mathbf{x}_t$ is the unobserved state and $\mathbf{z}_t$ is the observed process, while $\boldsymbol{\beta}_t$ and $\boldsymbol{\eta}_t$ are independent Brownian motions. A special case is when the term $\text{d} \boldsymbol{\eta}_t$ is discarded from the second equation (the observation equation). This concerns for instance Continuous-Time Auto-Regressive models. However, most books seem to consider only the case with measurement noise: see for instance Jazwinski, Øskendal or Sarkka and Solin.

Where can we find a description/discussion of the KF for the case with no measurement noise?

Yves
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  • Have you checked to see if the math still works if you simply set the covariance matrix of $\eta$ to zero? – conjectures May 30 '19 at 12:43
  • Yes the results could be obtained when this covariance, say $\mathbf{R}$, tends to zero. Yet It is not obvious since $\mathbf{R}^{-1}$ is used in the equations. See eqs (10.38) in Sarkka and Solin (freely available online). – Yves May 30 '19 at 15:43
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    Is it? In the recursion equations in Durbin & Koopman it is not. I imagine that there will be some considerations around initialisation. – conjectures May 30 '19 at 16:03
  • In Durbin & Koopman (sec 3.10 in my version) It seems that only discrete-time observations are considered, which makes a huge difference with the (ideal) continously oberved case. This corresponds to the limit of discrete-time obs. when the maximal step $t_{i+1} - t_i$ tends to zero. – Yves May 30 '19 at 16:27
  • Sure. I hadn't considered that. Nevertheless the intuition driving me here is that GPs without 'observation noise' are still GPs. – conjectures May 30 '19 at 16:35
  • Yes, I am pretty sure that differential equations still hold for the expectation and the covariance of the Gaussian distribution $\mathbf{x}_t \vert \mathcal{Z}_t$, and I hope to find them somewhere. – Yves May 30 '19 at 16:40
  • Did you ever get anywhere with this? I've been wondering if the computation can be speeded-up if we assume away observation noise. – Bob Aug 24 '20 at 10:33
  • Well I think I have a part of the answer: under general conditions on the SS model, when there is no measurement noise, the state vector $\mathbf{x}_t$ lies in the $\sigma$-field generated by the obsevations available at time $t$, i.e. by $\{y_s: \, s \leq t \}$. So the filtering covariance matrix is zero. – Yves Aug 24 '20 at 12:08

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