On partially observed jump diffusions I. The filtering equations

TL;DR

This paper extends filtering theory to partially observed jump diffusions driven by Wiener processes and Poisson measures, deriving equations for the evolution of conditional distributions under growth conditions.

Contribution

It introduces filtering equations for jump diffusions with coefficients satisfying growth conditions, extending diffusion filtering results to jump processes.

Findings

Derived equations for conditional distributions of jump diffusions.

Extended diffusion filtering results to jump processes.

Provided theoretical framework for filtering with jump diffusions.

Abstract

This paper is the first part of a series of papers on filtering for partially observed jump diffusions satisfying a stochastic differential equation driven by Wiener processes and Poisson martingale measures. The coefficients of the equation only satisfy appropriate growth conditions. Some results in filtering theory of diffusion processes are extended to jump diffusions and equations for the time evolution of the conditional distribution and the unnormalised conditional distribution of the unobserved process at time $t$, given the observations until $t$, are presented.