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Consistent parameter estimation for partially observed diffusions with small noise

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Abstract

In this paper we provide a consistency result for the MLE for partially observed diffusion processes with small noise intensities. We prove that if the underlying deterministic system enjoys an identifiability property, then any MLE is close to the true parameter if the noise intensities are small enough. The proof uses large deviations limits obtained by PDE vanishing viscosity methods. A deterministic method of parameter estimation is formulated. We also specialize our results to a binary detection problem, and compare deterministic and stochastic notions of identifiability.

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References

  1. Baras, J. S., and La Vigna, A. (1987), Real-time sequential hypothesis testing for diffusion signals, Proc. 26th IEEE CDC, Los Angeles, 1987, pp. 1153–1157, IEEE Press, New York.

    Google Scholar 

  2. Barles, G., and Perthame, B. (1987), Discontinuous solutions of deterministic optimal stopping time problems, Modél. Math. Anal. Numér. 21:557–579.

    Google Scholar 

  3. Campillo, F., and Le Gland, F. (1989), MLE for partially observed diffusions: direct maximization vs. the EM algorithm, Stochastic Process. Appl. 33:245–274.

    Google Scholar 

  4. Clark, J. M. C. (1978), The design of robust approximations to the stochastic differential equations of non-linear filtering, in: Communication Systems and Random Processes Theory (ed. J. K. Skwirzynski), pp. 721–734, Sijthoff and Nordhoff, Alphen aan den Rijn.

  5. Crandall, M. G., Evans, L. C., and Lions, P. L. (1984), Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 282:487–502.

    Google Scholar 

  6. Crandall, M. G., and Lions, P. L. (1983), Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277:1–42.

    Google Scholar 

  7. Davis, M. H. A. (1980), On a multiplicative functional transformation arising in non-linear filtering theory, Z. Wahrsch. Verw. Gebiete 54:125–139.

    Google Scholar 

  8. Dembo, A., and Zeitouni, O. (1986), Parameter estimation of partially observed continuous-time stochastic processes via the EM algorithm, Stochastic Process. Appl. 23:91–113.

    Google Scholar 

  9. Elliott, R. J. (1982), Stochastic Calculus and Applications, Springer-Verlag, New York.

    Google Scholar 

  10. Evans, L. C., and Ishii, H. (1985), A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities, Ann. Inst. H. Poincaré Anal. Non Linéaire 2:1–20.

    Google Scholar 

  11. Fleming, W. H., and Mitter, S. K. (1982), Optimal control and nonlinear filtering for nondegenerate diffusion processes, Stochastics 8:63–77.

    Google Scholar 

  12. Freidlin, M. I., and Wentzell, A. D. (1984), Random Perturbations of Dynamical Systems, Springer-Verlag, New York.

    Google Scholar 

  13. Friedman, A. V. (1964), Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ.

    Google Scholar 

  14. Hijab, O. (1984), Asymptotic Bayesian estimation of a first order equation with small diffusion, Ann. Probab. 12:890–902.

    Google Scholar 

  15. James, M. R. (1988), Asymptotic Nonlinear Filtering and Large Deviations with Application to Observer Design, Ph.D. Dissertation, University of Maryland (SRC Technical Report PhD-88-1, Systems Research Center).

  16. James, M. R. (1991), Finite time observer désign by probabilistic-variational methods, SIAM J. Control Optim. 29:954–967.

    Google Scholar 

  17. James, M. R., and Baras, J. S. (1988), Nonlinear filtering and large deviations: a PDE-control theoretic approach, Stochastics 23:391–412.

    Google Scholar 

  18. Pardoux, E. (1991), Filtrage non-linéaire et équations aux dérivées partielles stochastiques associées, in: Ecole d'Eté de Probabilités de Saint-Flour XIX, 1989 (ed. P. L. Hennequin), pp. 69–163 (LNM, 1464), Springer-Verlag, New York.

    Google Scholar 

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Authors and Affiliations

  1. Department of Systems Engineering, Australian National University, 0200, Canberra, ACT, Australia

    M. R. James

  2. Institut de Recherche en Informatique et Systèmes Aléatoires, Institut National de Recherche en Informatique et en Automatique, Campus de Beaulieu, 35042, Rennes Cédex, France

    F. Le Gland

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  1. M. R. James
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  2. F. Le Gland
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Additional information

This research was supported: by Systems Research Center, University of Maryland through NSF Grant CDR-85-00108 and AFOSR-URI Grant 87-0073; by Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, under ARO/MIT Grant DAAL-03-86-K-0171; by INRIA Sophia Antipolis, under ERO/INRIA Grant DAJA45-90-C-0008, and by the CNRS-GRAutomatique.

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James, M.R., Le Gland, F. Consistent parameter estimation for partially observed diffusions with small noise. Appl Math Optim 32, 47–72 (1995). https://doi.org/10.1007/BF01189903

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