self studying Oksendal book, on the Kalman filter chapter p.99, there is the following example
On the system $$dX_t=0,\text{ i.e. }X_t=X_0;E[X_0]=\hat{X}_0,E[X_0^2]=a^2$$ with observations $$dZ_t=X_tdt+mdV_t; Z_0=0$$
$dV_T$ being a Wiener process. Solving the corresponding Ricatti equation leads to the following equation for best linear estimator of $X_t:$ $$d\hat{X}_t=-\frac{a^2}{m^2+a^2t}\hat{X}_tdt+\frac{a^2}{m^2+a^2t}dZ_t\text{, }\hat{X}_0=0$$ or $$d\left(\hat{X}_t\exp\left(\int_0^t\frac{a^2}{m^2+a^2s}ds\right)\right)=\exp\left(\int_0^t\frac{a^2}{m^2+a^2s}ds\right)\frac{a^2}{m^2+a^2t}dZ_t$$ I can reach that point, however the book then gives the final expression for $\hat{X}_t$ as a function of $\hat{X}_0$ and $Z_t$ that I do not seem to be able to derive.
I can see that the lhs is the differential of $x * f(t)$ with $x=\hat{X}_t$ and $f(t)=\exp\left(\int_0^t\frac{a^2}{m^2+a^2s}ds\right)$, and I could use Ito rule to derive an expression but intuitively that would lead me nowhere...I would appreciate a nudge in the good direction ?
Thanks.