Estimation in Interacting Diffusions: Continuous and Discrete Sampling

Author(s)
Jaya Prakash Narayan Bishwal

ABSTRACT

Consistency and asymptotic normality of the sieve estimator and an approximate maximum likelihood estimator of the drift coefficient of an interacting particles of diffusions are studied. For the sieve estimator, observations are taken on a fixed time interval [0,*T*] and asymptotics are studied as the number of interacting particles increases with the dimension of the sieve. For the approximate maximum likelihood estimator, discrete observations are taken in a time interval [0,*T*] and asymptotics are studied as the number of interacting particles increases with the number of observation time points.

Consistency and asymptotic normality of the sieve estimator and an approximate maximum likelihood estimator of the drift coefficient of an interacting particles of diffusions are studied. For the sieve estimator, observations are taken on a fixed time interval [0,

KEYWORDS

Stochastic Differential Equations, Mean-Field Model, Large Interacting Systems, Diffusion Process, Discrete Observations, Approximate Maximum Likelihood Estimation, Sieve Estimation

Stochastic Differential Equations, Mean-Field Model, Large Interacting Systems, Diffusion Process, Discrete Observations, Approximate Maximum Likelihood Estimation, Sieve Estimation

Cite this paper

nullJ. Bishwal, "Estimation in Interacting Diffusions: Continuous and Discrete Sampling,"*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1154-1158. doi: 10.4236/am.2011.29160.

nullJ. Bishwal, "Estimation in Interacting Diffusions: Continuous and Discrete Sampling,"

References

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[2] D. Dawson, “Critical Dynamics and Fluctuations for a Mean-Field Model of Cooperative Behavior,” Journal of Statistical Physics, Vol. 31, No. 1, 1983, pp. 29-85. doi:10.1007/BF01010922

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[5] U. Grenander, “Abstract Inference,” Wiley, New York, 1981.

[6] H. T. Nguyen and T. D. Pham, “Identification of Nonstationary Diffusion Model by the Method of Sieves,” SIAM Journal of Control and Optimization, Vol. 20, No. 5, 1982, pp. 603-611. doi:10.1137/0320045

[7] R. A. Kasonga, “Maximum Likelihood Theory for Large Interacting Systems,” SIAM Journal on Applied Mathematics, Vol. 50, No. 3, 1990, pp. 865-875. doi:10.1137/0150050

[1] J. P. N. Bishwal, “Parameter Estimation in Stochastic Differential Equations,” Lecture Notes in Mathematics, Vol. 1923, Springer-Verlag, Berlin Hiedelberg, 2008.

[2] D. Dawson, “Critical Dynamics and Fluctuations for a Mean-Field Model of Cooperative Behavior,” Journal of Statistical Physics, Vol. 31, No. 1, 1983, pp. 29-85. doi:10.1007/BF01010922

[3] T. Ligget, “Interacting Particle Systems,” Springer-Ver- lag, New York, 1985.

[4] R. Carmona, J. P. Fouque and D. Vestal, “Interacting Particle Systems for the Computation of Rare Credit Portfolio Losses,” Finance and Stochastics, Vol. 13, No. 4, 2009, pp. 613-633. doi:10.1007/s00780-009-0098-8

[5] U. Grenander, “Abstract Inference,” Wiley, New York, 1981.

[6] H. T. Nguyen and T. D. Pham, “Identification of Nonstationary Diffusion Model by the Method of Sieves,” SIAM Journal of Control and Optimization, Vol. 20, No. 5, 1982, pp. 603-611. doi:10.1137/0320045

[7] R. A. Kasonga, “Maximum Likelihood Theory for Large Interacting Systems,” SIAM Journal on Applied Mathematics, Vol. 50, No. 3, 1990, pp. 865-875. doi:10.1137/0150050