S) \nu
\xi(t) + (\sigma_- \otimes S^\ast) \nu^\ast \xi^\ast(t) ) dt .
\label{eq:innovation-matrix}$$ The process $W(t)$ defined by (\[eq:innovation-matrix\]) is a $I \otimes \mathscr{Y}_t$ Wiener process with respect to $\vert \Sigma \rangle$ and is called the *innovations process*.

We follow the characteristic function method [@HSM05], [@VPB93], [@VPB92], whereby we postulate that the filter has the form $$d \tilde \pi_t( A \otimes X) = \mathcal{F}_t(A \otimes X) dt + \mathcal{H}_t(A
\otimes X) I\otimes dY(t) ,$$ where $\mathcal{F}_t$ and $\mathcal{H}_t$ are to be determined.

Let $f$ be square integrable, and define a process $c_f$ by $$dc_f(t) = f(t) c_f(t) dY(t), \ \ c_f(0)=1.$$ Then $I \otimes c_f(t)$ is adapted to $I \otimes \mathscr{Y}_t$, and the defining relation (\[eq:c-exp-def-matrix\]) implies that $$\mathbb{E}_\Sigma [ A \otimes (X(t) c_f(t) ) ] = \mathbb{E}_\Sigma [ \tilde \pi_t( A
\otimes X) I \otimes c_f(t) ) ]$$ holds for all $f$. By calculating the differentials of both sides, taking expectations and conditioning we obtain $$\begin{aligned}
\mathbb{E}_\Sigma [ A \otimes (dX(t) c_f(t)) ] &=& \mathbb{E}_\Sigma [ (I
\otimes c_f(t) ) \tilde \pi_t( \mathcal{G}(A \otimes X) ) \\
&& + (I \otimes f(t) c_f(t) ) \{\tilde  \pi_t( A \otimes (XL+L^\ast X) )  \notag \\
&& + \tilde \pi_t( A \sigma_+ 