scholarly journals Generalized Girsanov Transform of Processes and Zakai Equation with Jumps

2021 ◽  
Vol 2 (3) ◽  
Author(s):  
Masatoshi Fujisaki ◽  
Takashi Komatsu
Keyword(s):  
Zakai Equation ◽  
Girsanov Transform

scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

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.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

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.

A Zakai equation derivation of the extended Kalman filter

Automatica ◽  
2010 ◽  
Vol 46 (3) ◽  
pp. 620-624 ◽  
Author(s):  
Robert J. Elliott ◽  
Simon Haykin
Keyword(s):  
Kalman Filter ◽  
Extended Kalman Filter ◽  
Zakai Equation

scholarly journals On the Distribution of First Exit Time for Brownian Motion with Double Linear Time-Dependent Barriers

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Lin Xu ◽  
Dongjin Zhu
Keyword(s):  
Brownian Motion ◽  
Linear Time ◽  
Exit Time ◽  
Time Dependent ◽  
First Exit Time ◽  
Main Method ◽  
Analytical Expression ◽  
Finite Series ◽  
Girsanov Transform ◽  
Linear Barrier

This paper focuses on the first exit time for a Brownian motion with a double linear time-dependent barrier specified by y=a+bt, y=ct, (a>0, b<0, c>0). We are concerned in this paper with the distribution of the Brownian motion hitting the upper barrier before hitting the lower linear barrier. The main method we applied here is the Girsanov transform formula. As a result, we expressed the density of such exit time in terms of a finite series. This result principally provides us an analytical expression for the distribution of the aforementioned exit time and an easy way to compute the distribution of first exit time numerically.

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