# Accession Number:

## AD0741890

# Title:

## Etude de L'Existence de la Solution D'une Equation Integro-Differentielle Intervenant en Filtrage Statistique Non-Lineaire (The Study of the Existence of a Solution for an Integro-Differential Equation which Arises in Non-Linear Statistical Filtering),

# Descriptive Note:

# Corporate Author:

## ECOLE NATIONALE SUPERIEURE DES MINES DE PARIS FONTAINEBLEAU (FRANCE) CENTRE D'AUTOMATIQUE

# Personal Author(s):

# Report Date:

## 1971-12-01

# Pagination or Media Count:

## 33.0

# Abstract:

The work is a study of the parabolic integro-differential equation satisfied by the conditional probability density of a Ito stochastic process, given noisy information, when neither the process equation nor the observation are linear. This equation was established by Kushner, but no study of the existence of a solution has been made so far. The classical parabolic equation of diffusion is studied first, and the functional spaces in which its solution is defined described. Hypothesis are presented that allow to establish strong regularity results on this solution. Using this regularity, and some additional hypothesis, the original equation is transformed into an integral equation. It is then proved that with the hypothesis made, this equation does have a unique solution on any finite time interval, and that this solution possesses the properties of a probability density. Author

# Descriptors:

# Subject Categories:

- Statistics and Probability