A Forward Interpolation Equation of a Semimartingale by Observations over a Point Process

Abstract

Let $(\Omega ,\mathcal{F}_\infty ,{\bf P})$ be a complete probability space, and let $(\mathcal{F}_t )$, $t \in {\bf 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 $(\Omega ,\mathcal{F}_\infty ,{\bf P})$, where $\theta _t $ is an $(\mathcal{F}_t )$-adapted, $0 \leq t < \infty $, unobservable component and $(T_n ,X_n )$, $n \geq 1$, 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.