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Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs

  • Joscha DiehlEmail author
  • Peter K. Friz
  • Hilmar Mai
  • Harald Oberhauser
  • Sebastian Riedel
  • Wilhelm Stannat
Chapter
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 102)

Abstract

We consider complex stochastic systems in continuous time and space where the objects of interest are modelled via stochastic differential equations, in general high dimensional and with nonlinear coefficients. The extraction of quantifiable information from such systems has a long history and many aspects. We shall focus here on the perhaps most classical problems in this context: the filtering problem for nonlinear diffusions and the problem of parameter estimation, also for nonlinear and multidimensional diffusions. More specifically, we return to the question of robustness, first raised in the filtering community in the mid-1970s: will it be true that the conditional expectation of some observable of the signal process, given an observation (sample) path, depends continuously on the latter? Sadly, the answer here is no, as simple counterexamples show. Clearly, this is an unhappy state of affairs for users who effectively face an ill-posed situation: close observations may lead to vastly different predictions. A similar question can be asked in the context of (maximum likelihood) parameter estimation for diffusions. Some (apparently novel) counter examples show that, here again, the answer is no. Our contribution (Crisan et al., Ann Appl Probab 23(5):2139–2160, 2013); Diehl et al., A Levy-area between Brownian motion and rough paths with applications to robust non-linear filtering and RPDEs (2013, arXiv:1301.3799; Diehl et al., Pathwise stability of likelihood estimators for diffusions via rough paths (2013, arXiv:1311.1061) changed to yes, in other words: well-posedness is restored, provided one is willing or able to regard observations as rough paths in the sense of T. Lyons.

Keywords

Brownian Motion Stochastic Differential Equation Robust Version Iterate Integral Rough Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Joscha Diehl
    • 1
    Email author
  • Peter K. Friz
    • 1
    • 2
  • Hilmar Mai
    • 2
  • Harald Oberhauser
    • 3
  • Sebastian Riedel
    • 1
  • Wilhelm Stannat
    • 1
  1. 1.Technical University of BerlinBerlinGermany
  2. 2.Weierstrass InstituteBerlinGermany
  3. 3.University of OxfordOxfordUK

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