What is it about?

In this article we present a new methodology which gives estimators of the solution to Zakai’s equation which are unbiased, in the sense that they do not possess discretization error. More pre- cisely the Zakai equation can be expressed as the likelihood function associated to the non-linear filter in continuous time, or more alternatively the solution of a stochastic partial differential equation (SPDE).

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Why is it important?

In almost any numerical approximation of the Zakai equation (which is needed as it is not ana- lytically available) one is forced into a time discretization of the non-linear filter leading to discretization bias. We provide a technique for unbiasedness, which only has access to time-discretized numerical solutions and the estimate has finite variance. The approach is useful as it provides a ground truth for the true value of the solution. Moreover our method is embarrassingly parallel (in terms of implementation). The applications of the Zakai equation include model selection in statistics and the solution of a SPDE in applied mathematics.


You n this article, we get rid of the discretization bias associated with the Euler discretization of the solution to Zakai equation (the likelihood function).

Hamza Ruzayqat
King Abdullah University of Science and Technology

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This page is a summary of: Unbiased estimation of the solution to Zakai’s equation, Monte Carlo Methods and Applications, April 2020, De Gruyter, DOI: 10.1515/mcma-2020-2061.
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