Let $\left(\Omega, \mathcal{F}, \left\{\mathcal{F}_{t}\right\}_{t\geq 0}, P \right)$ be a complete probability space with a nondecreasing family of right continuous sub-$\sigma$-algebras $\mathcal{F}$. Let $h=(h_{t}), t \geq 0$ be integer right continuous Markov chain with values in the countable set ${0,1,2,...}$, this process is a pure birth process: \begin{equation*} \begin{cases} P(h_{t+\Delta t}-h_{t}=1| \mathcal{F}_{t})=a \cdot h_{t} \cdot \Delta t + o(\Delta t),\\ P(h_{t+\Delta t}-h_{t}=0| \mathcal{F}_{t})=1 - a \cdot h_{t} \cdot \Delta t + o(\Delta t), \end{cases} \end{equation*} where $\Delta t \rightarrow 0, \Delta t>0, a>0$. $h_{t}$ - a partially observable process, we know several its values: $h_{\tau_1}, h_{\tau_2}, ..., h_{\tau_n},$ where $0 \leq \tau_1 \leq \tau_2 \leq ... \leq \tau_n$ - the moments of jumps of the homogeneous Poisson process $\pi_{t}$ with parameter $\lambda>0$.
I want to build the optimal (in a mean square error sense) estimation of $h_{s}$ for $\forall s \in [0, t]$. Seems, it is $E(h_{s}|\sigma\{w:h_{\tau_1}, h_{\tau_2}, ..., h_{\tau_n}\})$. How can I do it?
I have tried to use the general equation of optimal nonlinear interpolation of partially observable random process (from this book Liptser, Shiryaev Statistics Of Random Processes):
$h_{t}$ can be represented as follows: $$h_{t}=h_{0}+\int_{0}^{t}{a \cdot h_{s}ds}+x_{t},$$ where $(x_{t},\mathcal{F}_{t})$ is a martingale.
Let $\xi_{t}$ is a process of the observation of $h_{t}$: $$\xi_{t}=\xi_{0}+\int_{0}^{t}{(h_{s}-\xi_{s-})d\pi_{s}}$$ $\pi_{t}$ can be represented as: $$\pi_{t}=\lambda \cdot t + W_{t},$$ where $W_{t}$ - wide-sense Wiener process. So $$h_{t}=\xi_{0}+\lambda \cdot \int_{0}^{t}{(h_{s}-\xi_{s-})d\pi_{s}}=\xi_{0}+\lambda \cdot \int_{0}^{t}{(h_{s}-\xi_{s-})d{s}}+\int_{0}^{t}{(h_{s}-\xi_{s-})d{W_{s}}}$$
The optimal estimation is $E(h_{s}|F_{t}^{\xi, \pi})$, where $F_{t}^{\xi, \pi}$ is the $\sigma$-algebra generated by $\xi=(\xi_{t})$ and $\pi=(\pi_{t})$.
The problem of such representation $\xi_{t}$ that the expression $(h_{t}-\xi_{t-})$ should be different from zero: $(h_{t}-\xi_{t-}) \geqslant C > 0$, because it is a denominator in the of $E(h_{s}|F_{t}^{\xi, \pi})$.