Continuity of the Filter with Unbounded Observation Coefficients

Abstract

In this article, we study the continuity with respect to the trajectories of the observation process for the filter associated with nonlinear filtering problems when the coefficients depend on both the signal and the observation and the observation coefficient is unbounded.

To achieve this task we define a formal unnormalized filter and prove by limiting arguments that it is related to the original filter through a generalized Bayes formula, and is locally Lipschitz continuous with respect to the uniform norm.

Keywords:

• Bayes formula
• Nonlinear filtering
• Pathwise continuity

Mathematics Subject Classification:

• 60G35
• 60H10
• 60G57

Acknowledgments

Partially supported by MURST project “Interazione e dipendenza nei modelli stocastici”—Anno 2008 - prot. C26A08H8YM Sapienza Università di Roma.