# Continuous-time Kalman filter with no observation/measurement noise

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?

• Have you checked to see if the math still works if you simply set the covariance matrix of $\eta$ to zero? – conjectures May 30 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 at 15:43
• 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 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 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 at 16:35