Solving the Zakai equation by ito's Method

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

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Model Problem for Integro-Differential Zakai Equation with Discontinuous Observation Processes

Applied Mathematics & Optimization ◽

10.1007/s00245-010-9131-8 ◽

2011 ◽

Vol 64 (1) ◽

pp. 37-69 ◽

Cited By ~ 1

Author(s):

R. Mikulevicius ◽

H. Pragarauskas

Keyword(s):

Model Problem ◽

Zakai Equation

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Real time solution of Duncan-Mortensen-Zakai equation without memory

2008 47th IEEE Conference on Decision and Control ◽

10.1109/cdc.2008.4739506 ◽

2008 ◽

Cited By ~ 1

Author(s):

Stephen S.-T. Yau ◽

Shing-Tung Yau

Keyword(s):

Real Time ◽

Zakai Equation ◽

Time Solution

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Fractional generalizations of filtering problems and their associated fractional Zakai equations

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-014-0197-x ◽

2014 ◽

Vol 17 (3) ◽

Author(s):

Sabir Umarov ◽

Frederick Daum ◽

Kenric Nelson

Keyword(s):

Brownian Motion ◽

Fractional Derivative ◽

Lévy Process ◽

Nonlinear Filtering ◽

Levy Process ◽

Zakai Equation ◽

Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

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