Abstract
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).
Similar content being viewed by others
References
Aldous, D.J., Eagleson, G.K.: On mixing and stability of limit theorems. Ann. Probab. 6, 325–331 (1978)
Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer, Berlin (2009)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)
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)
Fukasawa, M.: Realized volatility with stochastic sampling. Stoch. Process. Appl. 120, 829–852 (2010)
Gobet, E.: Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli 7, 899–912 (2001)
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)
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)
Jacod, J., Protter, P.: Asymptotic error distributions for the Euler method for stochastic differential equations. Ann. Probabil. 26, 267–307 (1998)
Jacod, J., Shiryaev, A.: Limit Theorems for Stochastics Processes, 2nd edn. Springer, Berlin (2003)
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Heidelberg (1992)
Kunita, H.: Nonlinear Filtering Problems I: Bayes Formulas and Innovations, The Oxford Handbook of Nonlinear Filtering, pp. 19–54. Oxford University Press, Oxford (2011)
Liptser, R.S., Shiryaev, A.N.: Statistics of Random Processes I. General Theory, 2nd edn. Springer, Berlin (2001)
Milstein, G.N., Tretyakov, M.V.: Monte Carlo methods for backward equations in nonlinear filtering. Adv. Appl. Probab. 41, 63–100 (2009)
Ogihara, T., Yoshida, N.: Quasi-likelihood analysis for nonsynchronously observed diffusion processes. Stoch. Process. Appl. 124, 2954–3008 (2014)
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)
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)
Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by JSPS KAKENHI Grant No 15K21598.
About this article
Cite this article
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). https://doi.org/10.1007/s13160-020-00411-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13160-020-00411-5