Filtering on manifolds
type: Item
instance of: scholarly article
title: Filtering on manifolds
zbMATH Open document ID: 0815.60054
DOI: 10.1007/BF00994916
author: István Gyöngy
published in: Acta Applicandae Mathematicae
publication date: 1995-06-30T00:00:00Z
review text: The main object of the article is the following system of Stratonovich stochastic differential equations \[ \begin{aligned} dx_ t & = b(t,x_ t,y_ t)dt + \sigma^ i(t,x_ t,y_ t)\circ dW^ i_ t + \overline{\sigma}^ j(t,x_ t) \circ dV^ j_ t,\\ dy_ t & = B(t,x_ t,y_ t)dt + dV_ t,\\ x_ 0 & = \xi,\quad y_ 0 \in \mathbb{R}^ d,\end{aligned} \] where \((W,V)\) is a \((d_ 1 + d_ 2)\)- dimensional Wiener process, \(b(t,\cdot,y)\), \(\sigma^ i(t,\cdot,y)\), \(\overline{\sigma}^ j(t,\cdot,y)\) are smooth vector fields, such that the solution \(x\) lies in the submanifold \(M\) of \(\mathbb{R}^{d_ 1}\), and \(\xi_ 0\) is a random vector in \(M\). The author considers the conditional distribution of \(x_ t\) under given \(\{y_ s; s\in [0;t]\}\). To describe this distribution the author considers a certain vector field on \(M\). He constructs a variant of the stochastic calculus in which the derivative is a derivative with respect to this field. The corresponding partial differential equations are considered. The main result is the following. For special \(b\), \(\sigma\), \(\overline{\sigma}\) the conditional distribution of \(x\) has a density with respect to the initial distribution \(\xi_ 0\) and this density satisfies the differential equation in the sense of the constructed stochastic calculus.
reviewed by: A. Ya. Dorogovtsev
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 93E11
zbMATH DE Number: 627456
zbMATH Keywords: Stratonovich stochastic differential equations
zbMATH Keywords: Wiener process
zbMATH Keywords: stochastic calculus
MaRDI profile type: Publication
cites work: Some filtering results in Riemann manifolds
cites work: On stochastic partial differential equations with unbounded coefficients
cites work: Stochastic partial differential equations on manifolds. I
cites work: Filtrage non lineaire avec observation sur une variete
description: scientific article; zbMATH DE number 627456