A forward interpolation equation of a semimartingale by observations over a point process

Let $(\Omega,\mathcal{F}_\infty,\mathbf{P})$ be a complete probability space, and let $(\mathcal{F}_t )$, $t\in\mathbf{R}_ + $, be a nondecreasing right-continuous family of sub-$\sigma $-algebras of $\mathcal{F}_\infty$ completed by sets from $\mathcal{F}_\infty$ of zero probability. A two-dimensional partially observable stochastic process is given on the probability space $(\Omega,\mathcal{F}_\infty,\mathbf{P})$, where $\theta _t $ is an $(\mathcal{F}_t )$-adapted, $0\leq t<\infty$, unobservable component and $(T_n ,X_n)$, $n \ge 1$, is an observable one. We consider the problem of optimal interpolation, which consists of finding an optimal mean square estimate $\theta_s$ from the observations of the process $(T_n,X_n)$ on $[0,t]$, $t\geq s$. This paper contains a deduction of an equation of optimal nonlinear interpolation on the basis of an equation of optimal nonlinear filtering.