Abstract.
The paper is concerned with completing “unfinished business” on a robust representation formula for the conditional expectation operator of nonlinear filtering. Such a formula, robust in the sense that its dependence on the process of observations is continuous, was stated in [2] without proof. The main purpose of this paper is to repair this deficiency.
The formula is “almost obvious” as it can be derived at a formal level by a process of integration-by-parts applied to the stochastic integrals that appear in the integral representation formula. However, the rigorous justification of the formula is quite subtle, as it hinges on a measurability argument the necessity of which is easy to miss at first glance. The continuity of the representation (but not its validity) was proved by Kushner [9] for a class of diffusions.
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Here we follow the definition given in [11].
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Clark, J., Crisan, D. On a robust version of the integral representation formula of nonlinear filtering. Probab. Theory Relat. Fields 133, 43–56 (2005). https://doi.org/10.1007/s00440-004-0412-5
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DOI: https://doi.org/10.1007/s00440-004-0412-5