Comparison of linear and nonlinear filtration of some Markov processes

Let ($\theta_t,\eta_t$) be a two-dimensional Markov process where $\theta_t$ is a homogeneous Markov process with two states and $\eta_t$ satisfies equation (1).
We solve the problem of filtration (linear and nonlinear) of the process $\theta_t$ under the condition that it is possible to observe $\eta_t$. The results of comparison of the linear and nonlinear filtration are given.