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On a robust version of the integral representation formula of nonlinear filtering
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  • Published: 10 February 2005

On a robust version of the integral representation formula of nonlinear filtering

  • J.M.C. Clark1 &
  • D. Crisan2 

Probability Theory and Related Fields volume 133, pages 43–56 (2005)Cite this article

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  • 18 Citations

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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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Author information

Authors and Affiliations

  1. Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London, SW7 2BT, United kingdom

    J.M.C. Clark

  2. Department of Mathematics, Imperial College London, 180 Queen’s Gate, London, SW7 2BZ, United kingdom

    D. Crisan

Authors
  1. J.M.C. Clark
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  2. D. Crisan
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Correspondence to J.M.C. Clark.

Additional information

Here we follow the definition given in [11].

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Cite this article

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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  • Received: 14 August 2004

  • Revised: 16 November 2004

  • Published: 10 February 2005

  • Issue Date: September 2005

  • DOI: https://doi.org/10.1007/s00440-004-0412-5

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Keywords

  • Stochastic Process
  • Probability Theory
  • Integral Representation
  • Mathematical Biology
  • Conditional Expectation
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