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Approximate martingale estimating functions for stochastic differential equations with small noises. (English) Zbl 1145.62065

Summary: An approximate martingale estimating function with an eigenfunction is proposed for an estimation problem about an unknown drift parameter for a one-dimensional diffusion process with small perturbed parameter \(\epsilon \) from discrete time observations at \(n\) regularly spaced time points \(k/n, k=0,1,\cdots ,n\). We show the asymptotic efficiency of an \(M\)-estimator derived from the approximate martingale estimating function as \(\varepsilon \rightarrow 0\) and \(n\rightarrow \infty \) simultaneously.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
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[1] Azencott, R., Formule de Taylor stochastique et développement asymptotique d’intégrales de Feynmann, (Séminaire de Probabilités XVI; Supplément: Géométrie Différentielle Stochastique. Séminaire de Probabilités XVI; Supplément: Géométrie Différentielle Stochastique, Lecture Notes In Math, vol. 921 (1982), Springer Verlag: Springer Verlag Berlin), 237-285 · Zbl 0484.60064
[2] Bibby, B. M.; Sørensen, M., Martingale estimating functions for discretely observed diffusion processes, Bernoulli, 1, 17-39 (1995) · Zbl 0830.62075
[3] Bibby, B. M.; Sørensen, M., On estimation for discretely observed diffusions: A review, Theory Stoch. Process., 2, 49-56 (1996) · Zbl 0893.62082
[5] Florens-Zmirou, D., Approximate discrete time schemes for statistics of diffusion processes, Statistics, 20, 547-557 (1989) · Zbl 0704.62072
[6] Freidlin, M. I.; Wentzell, A. D., Random Perturbations of Dynamical Systems (1998), Springer-Verlag: Springer-Verlag New York · Zbl 0922.60006
[7] Genon-Catalot, V., Maximum contrast estimation for diffusion processes from discrete observations, Statistics, 21, 99-116 (1990) · Zbl 0721.62082
[8] Genon-Catalot, V.; Jacod, J., On the estimation of the diffusion coefficient for multidimensional diffusion processes, Ann. Inst. H. Poincaré Probab. Statist., 29, 119-151 (1993) · Zbl 0770.62070
[10] Ibragimov, I. A.; Has’minskii, R. Z., Statistical Estimation (1981), Springer-Verlag: Springer-Verlag New York
[11] Kessler, M., Estimation of an ergodic diffusion from discrete observations, Scand. J. Statist., 24, 211-229 (1997) · Zbl 0879.60058
[12] Kessler, M.; Sørensen, M., Estimating equations based on eigenfunctions for a discretely observed diffusion process, Bernoulli, 5, 299-314 (1999) · Zbl 0980.62074
[13] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0925.65261
[14] Kunitomo, N.; Takahashi, A., The asymptotic expansion approach to the valuation of interest rate contingent claims, Math. Finance, 11, 117-151 (2001) · Zbl 0994.91023
[15] Kutoyants, Yu. A., (Prakasa Rao, B. L.S., Parameter Estimation for Stochastic Processes (1984), Heldermann: Heldermann Berlin) · Zbl 0542.62073
[16] Kutoyants, Yu. A., Identification of Dynamical Systems with Small Noise (1994), Kluwer: Kluwer Dordrecht · Zbl 0831.62058
[17] Laredo, C. F., A sufficient condition for asymptotic sufficiency of incomplete observations of a diffusion process, Ann. Statist., 18, 1158-1171 (1990) · Zbl 0725.62073
[18] Sakamoto, Y.; Yoshida, N., Asymptotic expansion formulas for functionals of \(\epsilon \)-Markov processes with a mixing property, Ann. Inst. Statist. Math., 56, 545-597 (2004) · Zbl 1063.62017
[19] Sørensen, M., Estimating functions for discretely observed diffusions: A review, (Basawa, I. V.; Godambe, V. P.; Taylor, R. L., Selected Proceedings of the Symposium on Estimating Functions. Selected Proceedings of the Symposium on Estimating Functions, IMS Lecture Notes-Monograph Series, vol. 32 (1997), Institute of Mathematical Statistics: Institute of Mathematical Statistics Hayward), 305-325 · Zbl 0906.62079
[21] Sørensen, M.; Uchida, M., Small diffusion asymptotics for discretely sampled stochastic differential equations, Bernoulli, 9, 1051-1069 (2003) · Zbl 1043.60050
[22] Takahashi, A.; Yoshida, N., An asymptotic expansion scheme for optimal investment problems, Stat. Inference Stoch. Process., 7, 153-188 (2004) · Zbl 1181.91302
[23] Uchida, M., Estimation for discretely observed small diffusions based on approximate martingale estimating functions, Scand. J. Statist., 31, 553-566 (2004) · Zbl 1062.62162
[24] Uchida, M., Martingale estimating functions based on eigenfunctions for discretely observed small diffusions, Bull. Inform. Cybernet., 38, 1-13 (2006) · Zbl 1270.62117
[25] Uchida, M.; Yoshida, N., Information criteria for small diffusions via the theory of Malliavin-Watanabe, Statist. Inference Stoch. Process., 7, 35-67 (2004) · Zbl 1333.62032
[26] Uchida, M.; Yoshida, N., Asymptotic expansion for small diffusions applied to option pricing, Statist. Inference. Stoch. Process., 7, 189-223 (2004) · Zbl 1125.62311
[27] Yoshida, N., Asymptotic behavior of \(M\)-estimator and related random field for diffusion process, Ann. Inst. Statist. Math., 42, 221-251 (1990) · Zbl 0723.62048
[28] Yoshida, N., Asymptotic expansion of maximum likelihood estimators for small diffusions via the theory of Malliavin-Watanabe, Probab. Theory Related Fields, 92, 275-311 (1992) · Zbl 0767.60035
[29] Yoshida, N., Asymptotic expansion for statistics related to small diffusions, J. Japan Statist. Soc., 22, 139-159 (1992) · Zbl 0778.62018
[30] Yoshida, N., Estimation for diffusion processes from discrete observation, J. Multivariate Anal., 41, 220-242 (1992) · Zbl 0811.62083
[31] Yoshida, N., Conditional expansions and their applications, Stochastic Process. Appl., 107, 53-81 (2003) · Zbl 1075.60515
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