Short Communications

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

N. V. Kvashko

M. V. Lomonosov Moscow State University

Abstract: 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.

Keywords: probability space, $\sigma $-algebra, point process, jump measure of a process, filtration of observations, martingale, semimartingale, drift, Dolé, ans measure, compensator, filtering, interpolation.

Received: 10.08.1992

 English version: Theory of Probability and its Applications, 1995, 40:1, 162–165

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