Robustness in Stochastic Filtering and Maximum Likelihood Estimation for SDEs

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 102)


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.


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.


  1. 1.
    Bagchi, A., Karandikar, R.: White noise theory of robust nonlinear filtering with correlated state and observation noises. Syst. Control Lett. 23(2), 137–148 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer, New York/London (2008)Google Scholar
  3. 3.
    Breiman, L.: Probability. Classics in Applied Mathematics. SIAM, Philadelphia (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cass, T., Lyons, T.: Evolving communities with individual preferences (2013, preprint). arXiv:1303.4243Google Scholar
  5. 5.
    Clark, J.M.C.: The design of robust approximations to the stochastic differential equations of nonlinear filtering. In: Communication Systems and Random Process Theory: Proceedings of 2nd NATO Advanced Study Institute, Darlington, 1977, pp. 721–734. NATO Advanced Study Institute Series, Series E, Applied sciences, vol. 25. Sijthoff & Noordhoff, Alphen aan den Rijn (1978)Google Scholar
  6. 6.
    Clark, J.M.C., Crisan, D.: On a robust version of the integral representation formula of nonlinear filtering. Probab. Theory Relat. F. 133(1), 43–56 (2005). doi: 10.1007/s00440-004-0412-5 CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Crisan, D., Diehl, J., Friz, P., Oberhauser, H.: Robust filtering: multdimensional noise and multidimensional observation. Ann. Appl. Probab. 23(5), 2139–2160 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Crisan, D., Rozovskiĭ, B.(eds.): The Oxford Handbook of Nonlinear Filtering, xiv, p. 1063. Oxford University Press, Oxford (2011)Google Scholar
  9. 9.
    Davis, M.: Pathwise nonlinear filtering. In: Stochastic Systems: The Mathematics of Filtering and Identification and Applications: Proceedings of the NATO Advanced Study Institute, Les Arcs, 1980, pp. 505–528 (1981)CrossRefGoogle Scholar
  10. 10.
    Davis, M.: Pathwise nonlinear filtering with correlated noise. In: Crisan, D., Rozovskiĭ, B.L. (eds.) The Oxford Handbook of Nonlinear Filtering, pp. 403–424. Oxford University Press, Oxford (2011)Google Scholar
  11. 11.
    Davis, M.H.A.: On a multiplicative functional transformation arising in nonlinear filtering theory. Z. Wahrsch. Verw. Gebiete 54(2), 125–139 (1980). doi: 10.1007/BF00531444.
  12. 12.
    Davis, M.H.A.: A pathwise solution of the equations of nonlinear filtering. Teor. Veroyatnost. i Primenen. 27(1), 160–167 (1982)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Davis, M.H.A., Spathopoulos, M.P.: Pathwise nonlinear filtering for nondegenerate diffusions with noise correlation. SIAM J. Control Optim. 25(2), 260–278 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Diehl, J., Friz, P., Mai, H.: Pathwise stability of likelihood estimators for diffusions via rough paths (2013, preprint). arXiv:1311.1061Google Scholar
  15. 15.
    Diehl, J., Friz, P., Stannat, W.: Measure valued rough differential equations (2014, in preparation)Google Scholar
  16. 16.
    Diehl, J., Oberhauser, H., Riedel, S.: A Levy-area between Brownian motion and rough paths with applications to robust non-linear filtering and RPDEs (2013, preprint). arXiv:1301.3799Google Scholar
  17. 17.
    Doss, H.: Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. H. Poincaré Sect. B (N.S.) 13(2), 99–125 (1977)Google Scholar
  18. 18.
    Elliott, R., Kohlmann, M.: Robust filtering for correlated multidimensional observations. Math. Z. 178(4), 559–578 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Florchinger, P.: Zakai equation of nonlinear filtering with unbounded coefficients. the case of dependent noises. Syst. Control Lett.21(5), 413–422 (1993)Google Scholar
  20. 20.
    Florchinger, P., Nappo, G.: Continuity of the filter with unbounded observation coefficients. Stoch. Anal. Appl. 29(4), 612–630 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Friz, P.: Continuity of the Ito-map for Hoelder rough paths with applications to the support theorem in Hoelder norm. In: Probability and Partial Differential Equations in Modern Applied Mathematics. IMA Volumes in Mathematics and its Applications, vol. 140, pp. 117–135. Springer, New York (2005)Google Scholar
  22. 22.
    Friz, P., Riedel, S.: Integrability of (non-) linear rough differential equations and integrals. Stoch. Anal. Appl. 31(2), 336–358 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Friz, P.K., Victoir, N.B.: Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics, vol. 120. Cambridge University Press, Cambridge (2010)Google Scholar
  24. 24.
    Gyöngy, I.: On the approximation of stochastic partial differential equations i. Stoch. Stoch. Rep. 25(2), 59–85 (1988)zbMATHGoogle Scholar
  25. 25.
    Gyöngy, I.: On the approximation of stochastic partial differential equations ii. Stoch. Stoch. Rep. 26(3), 129–164 (1989)CrossRefzbMATHGoogle Scholar
  26. 26.
    Kushner, H.J.: A robust discrete state approximation to the optimal nonlinear filter for a diffusion. Stochastics 3(2), 75–83 (1979)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Lyons, T.J.: Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young. Math. Res. Lett. 1(4), 451–464 (1994)zbMATHMathSciNetGoogle Scholar
  28. 28.
    Lyons, T.J., Caruana, M., Lévy, T.: Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics, vol. 1908. Springer, Berlin (2007). Lectures from the 34th Summer School on Probability Theory, Saint-Flour, 6–24 July 2004Google Scholar
  29. 29.
    Lyons, T.J., Qian, Z.: System Control and Rough Paths. Oxford Mathematical Monographs. Oxford University Press, Oxford (2002)CrossRefzbMATHGoogle Scholar
  30. 30.
    Sussmann, H.J.: On the gap between deterministic and stochastic ordinary differential equations. Ann. Probab. 6(1), 19–41 (1978)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Technical University of BerlinBerlinGermany
  2. 2.Weierstrass InstituteBerlinGermany
  3. 3.University of OxfordOxfordUK

Personalised recommendations