Assume a linear stochastic dynamical system $$ \begin{align} \mathrm{d}x&=-J_{1,1}x+J_{1,2}y+\sigma_{1}\,\mathrm{d}W_{1}\\\mathrm{d}y&=-J_{2,1}x-J_{2,2}y-\sigma_{1}\,\mathrm{d}W_{1}+\sigma_{2}\,\mathrm{d}W_{2} \end{align} $$ with the $J_{k,l}>0$ and two independent standard Wiener processes $W_1$ and $W_2$. We're looking here at the linearization of a non-linear system, that fluctuates around its steady state.
Assume we have long-time observations of a time series $x_{t}$ (with negligible measurement error for now) and know the structure of the above dynamical system. However, we have no direct access on $y_{t}$ and the parameter values are not (or only partially) known.
Here's my question:
Which method allows to reconstruct the hidden time series $y_{t}$ from the observed data $x_{t}$ and the known structure of the SDE system? If reconstructing the time series is too strong, then I would at least like to have an estimate of its statistical properties.
I assume this is a standard problem, but I'm a beginner in time series analysis and struggle with the zoo of available methods.