# Time-discretization of the Zakai equation for diffusion processes observed in correlated noise

@article{Florchinger1991TimediscretizationOT, title={Time-discretization of the Zakai equation for diffusion processes observed in correlated noise}, author={Patrick Florchinger and François Le Gland}, journal={Lecture Notes in Control and Information Sciences}, year={1991}, volume={144}, pages={228-237} }

A time discretization scheme is provided for the Zakai equation, a stochastic PDE which gives the conditional law of a diffusion process observed in white-noise. The case where the observation noise and the state noise are correlated, is considered. The numerical scheme is based on a Trotter-like product formula, which exhibits prediction and correction steps, and for which an error estimate of order δ is proved, where δ is the time discretization step. The correction step is associated with a…

## 72 Citations

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It is shown that the rate of convergence is in general of order √ δ ( δ is the time step), but in the case when there is no correlation between the signal and the observation for the Zakai equation, the order of convergence becomes δ.

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A nonlinear model with observations driven by correlated Wiener processes and point processes with a splitting-up approximate solution is considered, which proves its half-order convergence to the exact solution of the Zakai equation.

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- Computer Science, MathematicsJournal of Scientific Computing
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This paper approximate the Zakai equation with two equations consisting of a first-order stochastic partial differential equation and a deterministic second-order partial differential equations for nonlinear filtering problems via solving their corresponding Zakai equations using the splitting-up technique.

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This paper presents two operator splitting-up approximations for the Kushner equation and it is shown that one of them is equivalent to the up approximation to the Zakai equation with normalization on each step.

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### Wiener Chaos and Nonlinear Filtering

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The paper discusses two algorithms for solving the Zakai equation in the time-homogeneous diffusion filtering model with possible correlation between the state process and the observation noise with main existing results about these Wiener chaos algorithms summarized.

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