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Asymptotic error distributions of the Euler method for continuous-time nonlinear filtering


In this paper, we deduce the asymptotic error distribution of the Euler method for the nonlinear filtering problem with continuous-time observations. As studied in previous works by several authors, the error structure of the method is characterized by conditional expectations of some functionals of multiple stochastic integrals. Our main result is to prove the stable convergence of a sequence of such conditional expectations by using the techniques of martingale limit theorems in the spirit of Jacod (On continuous conditional Gaussian martingales and stable convergence in law, seminaire de probabilites, XXXI, lecture notes in mathematics, Springer, Berlin, 1997).

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  1. Aldous, D.J., Eagleson, G.K.: On mixing and stability of limit theorems. Ann. Probab. 6, 325–331 (1978)

    MathSciNet  Article  Google Scholar 

  2. Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer, Berlin (2009)

    Book  Google Scholar 

  3. Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)

    Book  Google Scholar 

  4. Clément, D., Kohatsu-Higa, A., Lamberton, D.: A duality approach for the weak approximation of stochastic differential equations. Ann. Appl. Probab. 16, 1124–1154 (2006)

    MathSciNet  Article  Google Scholar 

  5. Fukasawa, M.: Realized volatility with stochastic sampling. Stoch. Process. Appl. 120, 829–852 (2010)

    MathSciNet  Article  Google Scholar 

  6. Gobet, E.: Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli 7, 899–912 (2001)

    MathSciNet  Article  Google Scholar 

  7. Jacod, J.: On Continuous Conditional Gaussian Martingales and Stable Convergence in Law, Seminaire de Probabilites, XXXI, Lecture Notes in Mathematics, vol. 1655, pp. 232–246. Springer, Berlin (1997)

    MATH  Google Scholar 

  8. Jacod, J., Li, Y., Mykland, P., Podolskij, P., Vetter, M.: Microstructure noise in the continuous case: the pre-averaging approach. Stoch. Process. Appl. 119, 2249–2276 (2009)

    MathSciNet  Article  Google Scholar 

  9. Jacod, J., Protter, P.: Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probabil. 26, 267–307 (1998)

    MathSciNet  Article  Google Scholar 

  10. Jacod, J., Shiryaev, A.: Limit Theorems for Stochastics Processes, 2nd edn. Springer, Berlin (2003)

    Book  Google Scholar 

  11. Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg (1992)

    Book  Google Scholar 

  12. Kunita, H.: Nonlinear Filtering Problems I: Bayes Formulas and Innovations, The Oxford Handbook of Nonlinear Filtering, pp. 19–54. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  13. Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes I. General Theory, 2nd edn. Springer, Berlin (2001)

    Book  Google Scholar 

  14. Milstein, G.N., Tretyakov, M.V.: Monte Carlo methods for backward equations in nonlinear filtering. Adv. Appl. Probab. 41, 63–100 (2009)

    MathSciNet  Article  Google Scholar 

  15. Ogihara, T., Yoshida, N.: Quasi-likelihood analysis for nonsynchronously observed diffusion processes. Stoch. Process. Appl. 124, 2954–3008 (2014)

    MathSciNet  Article  Google Scholar 

  16. Picard, J.: Approximation of Nonlinear Filtering Problems and Order of Convergence, Filtering and Control of Random Processes (Lecture Notes Control Inform. Sci. 61), pp. 219–236. Springer, Berlin (1984)

    Google Scholar 

  17. Talay, D.: Efficient Numerical Schemes for the Approximation of Expectations of Functionals of the Solution of a SDE and Applications. Filtering and Control of Random Processes (Lecture Notes Control Inform. Sci. 61), pp. 294–313. Springer, Berlin (1984)

    Google Scholar 

  18. Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

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Correspondence to Hideyuki Tanaka.

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Ogihara, T., Tanaka, H. Asymptotic error distributions of the Euler method for continuous-time nonlinear filtering. Japan J. Indust. Appl. Math. 37, 383–413 (2020).

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  • Nonlinear filtering
  • Euler method
  • Stable convergence
  • Martingale limit theorem

Mathematics Subject Classification

  • 60G35
  • 93E11
  • 60F05
  • 65C20