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Conditional expansions and their applications. (English) Zbl 1075.60515

Summary: We consider a conditional limit theorem and conditional asymptotic expansions. Our discussion will be based on the Malliavin calculus. First, we treat a problem of lifting limit theorems to their conditional counterparts. Next, we provide asymptotic expansions in a general setting including the so-called small \(\sigma \)-models. In order to give a basis to the asymptotic expansion scheme for perturbed jump systems, we will build an extension to the Watanabe theory in part. Finally, we derive the asymptotic expansions (double Edgeworth expansions) of conditional expectations.

MSC:

60F99 Limit theorems in probability theory
60H07 Stochastic calculus of variations and the Malliavin calculus
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60J60 Diffusion processes
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[1] Barndorff-Nielsen, O. E.; Cox, D. R., Inference and Asymptotics (1994), Chapman & Hall: Chapman & Hall London · Zbl 0826.62004
[2] Bichteler, K.; Gravereaux, J.-B.; Jacod, J., Malliavin Calculus for Processes with Jumps (1987), Gordon and Breach Science Publishers: Gordon and Breach Science Publishers New York, London, Paris, Montreux, Tokyo · Zbl 0706.60057
[3] Del Moral, P.; Jacod, J.; Protter, Ph., The Monte-Carlo method for filtering with discrete-time observations, Probab. Theory Relat. Fields, 120, 346-368 (2001) · Zbl 0979.62072
[4] Dermoune, A.; Kutoyants, Yu. A., Expansion of distribution function of maximum likelihood estimate for misspecified diffusion type observations, Stochastics and Stochastic Reports, 52, 121-145 (1995) · Zbl 0852.62075
[5] Genon-Catalot, V.; Jacod, J., On the estimation of the diffusion coefficient for multi-dimensional diffusion processes, Ann. Inst. Henri Poincaré, 29, 1, 119-151 (1993) · Zbl 0770.62070
[6] Ghosh, J. K., Higher order asymptotics (1994), IMS: IMS California · Zbl 1163.62305
[7] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes (1990), North-Holland/Kodansha: North-Holland/Kodansha Tokyo
[8] Jacod, J., 1996. On continuous conditional Gaussian martingales and stable convergence in law. Prépublication No. 339 du Laboratoire de Probabilités, Université Paris 6.; Jacod, J., 1996. On continuous conditional Gaussian martingales and stable convergence in law. Prépublication No. 339 du Laboratoire de Probabilités, Université Paris 6. · Zbl 0884.60038
[9] Kashiwakura, K., Yoshida, N., 2001. Asymptotic and hybrid expansions. In preparation.; Kashiwakura, K., Yoshida, N., 2001. Asymptotic and hybrid expansions. In preparation.
[10] Kim, Y. J.; Kunitomo, N., Pricing options under stochastic interest rates, Asia Pacific Financial Markets, 6, 49-70 (1999) · Zbl 1157.91363
[11] Kitagawa, G., Non-Gaussian state space modelling of non-stationary time series, J. Amer. Statist. Assoc., 82, 503-514 (1987) · Zbl 0679.62070
[12] Kunitomo, N., Takahashi, A., 1998. On validity of the asymptotic expansion approach in contingent claim analysis. Discussion Paper No.98-F-6, Faculty of Economics, University of Tokyo.; Kunitomo, N., Takahashi, A., 1998. On validity of the asymptotic expansion approach in contingent claim analysis. Discussion Paper No.98-F-6, Faculty of Economics, University of Tokyo.
[13] Kunitomo, N.; Takahashi, A., The aysmptotic expansion approach to the valuation of interest rate contingent claims, Math. Finance, 11, 117-151 (2001) · Zbl 0994.91023
[14] Kusuoka, S.; Stroock, D. W., Precise asymptotics of certain Wiener functionals, J. Funct. Anal., 99, 1-74 (1991) · Zbl 0738.60054
[15] Kutoyants, Yu. A., Identification of Dynamical Systems with Small Noise (1994), Kluwer: Kluwer Dordrecht · Zbl 0831.62058
[16] Malliavin, P., 1997. Stochastic Analysis. Springer, Berlin, Heidelberg, New York.; Malliavin, P., 1997. Stochastic Analysis. Springer, Berlin, Heidelberg, New York. · Zbl 0878.60001
[17] Masuda, H., Yoshida, N., 2002. The double Edgeworth expansion for the partial non-Gaussian state space models. Preprint; Masuda, H., Yoshida, N., 2002. The double Edgeworth expansion for the partial non-Gaussian state space models. Preprint
[18] Nualart, D., The Malliavin Calculus and Related Topics (1995), Springer: Springer Berlin · Zbl 0837.60050
[19] Picard, J., Efficiency of the extended Kalman filter for nonlinear systems with small noise, SIAM J. Appl. Math., 51, 843-885 (1991) · Zbl 0733.93075
[20] Sakamoto, Y.; Yoshida, N., Expansion of perturbed random variables based on generalized Wiener functionals, J. Multivariate Anal., 59, 1, 34-59 (1996) · Zbl 0866.60046
[21] Shephard, N., Partial non-Gaussian state space, Biometrika, 81, 115-131 (1994) · Zbl 0796.62079
[22] \(Sø\); \(Sø\)
[23] Sweeting, T. J., Asymptotic conditional inference for the offspring mean of a supercritical Galton-Watson process, Ann. Statist., 14, 925-933 (1986) · Zbl 0633.62084
[24] Takahashi, A., 1995. Essays on the valuation problems of contingent claims. Ph.D. Dissertation, Haas School of Business, University of California, Berkeley.; Takahashi, A., 1995. Essays on the valuation problems of contingent claims. Ph.D. Dissertation, Haas School of Business, University of California, Berkeley.
[25] Takahashi, A., An asymptotic expansion approach to pricing contingent claims, Asia-Pacific Financial Markets, 6, 115-151 (1999) · Zbl 1153.91568
[26] Takahashi, A., Yoshida, N., 2001. Asymptotic expansion scheme for optimal portfolio. Submitted for publication.; Takahashi, A., Yoshida, N., 2001. Asymptotic expansion scheme for optimal portfolio. Submitted for publication. · Zbl 1047.91538
[27] Takanobu, S., Watanabe, S., 1993. Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations. In: Elworthy, K.D., Ikeda, N. (Eds.), Asymptotic problems in probability theory: Wiener functionals and asymptotics. Proceedings of the Taniguchi International Symposium, Sanda and Kyoto, 1990, Longman, UK, pp. 194-241.; Takanobu, S., Watanabe, S., 1993. Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations. In: Elworthy, K.D., Ikeda, N. (Eds.), Asymptotic problems in probability theory: Wiener functionals and asymptotics. Proceedings of the Taniguchi International Symposium, Sanda and Kyoto, 1990, Longman, UK, pp. 194-241. · Zbl 0801.60020
[28] Uchida, M., Yoshida, N., 1999. Information criteria for small diffusions via the theory of Malliavin-Watanabe. Submitted for publication.; Uchida, M., Yoshida, N., 1999. Information criteria for small diffusions via the theory of Malliavin-Watanabe. Submitted for publication. · Zbl 1333.62032
[29] Watanabe, S., 1983. Malliavin’s calculus in terms of generalized Wiener functionals. In: Kallianpur, G. (Ed.), Theory and Applications of Random Fields, Proc. IFIP Conference Bangalore 1982, Lecture Notes in Control Information Science, Vol. 49. Springer, Berlin, Heidelberg, New York, pp. 284-290.; Watanabe, S., 1983. Malliavin’s calculus in terms of generalized Wiener functionals. In: Kallianpur, G. (Ed.), Theory and Applications of Random Fields, Proc. IFIP Conference Bangalore 1982, Lecture Notes in Control Information Science, Vol. 49. Springer, Berlin, Heidelberg, New York, pp. 284-290.
[30] Watanabe, S., 1984. Lectures on Stochastic Differential Equations and Malliavin Calculus. Tata Institute of Fundamental Research, Springer, Bombay.; Watanabe, S., 1984. Lectures on Stochastic Differential Equations and Malliavin Calculus. Tata Institute of Fundamental Research, Springer, Bombay. · Zbl 0546.60054
[31] Watanabe, S., Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab., 15, 1-39 (1987) · Zbl 0633.60077
[32] Yoshida, N., Asymptotic expansion for small diffusions via the theory of Malliavin-Watanabe, Prob. Theory Related Fields, 92, 275-311 (1992) · Zbl 0767.60035
[33] Yoshida, N., Asymptotic expansion for statistics related to small diffusions, J. Jpn. Statist. Soc., 22, 2, 139-159 (1992) · Zbl 0778.62018
[34] Yoshida, N., Asymptotic expansion of Bayes estimators for small diffusions, Probab. Theory Relat. Fields, 95, 429-450 (1993) · Zbl 0796.62071
[35] Yoshida, N., Asymptotic expansions for perturbed systems on Wiener spacemaximum likelihood estimators, J. Multivariate Anal., 57, 1-36 (1996) · Zbl 0845.62054
[36] Yoshida, N., 1999. Malliavin calculus and statistics. In: S. Kotz et. al. (Eds.), Encyclopedia of Statistical Sciences, Update Vol. 3. Wiley, New York.; Yoshida, N., 1999. Malliavin calculus and statistics. In: S. Kotz et. al. (Eds.), Encyclopedia of Statistical Sciences, Update Vol. 3. Wiley, New York.
[37] Yoshida, N., 2001. Conditional asymptotics. Preprint.; Yoshida, N., 2001. Conditional asymptotics. Preprint.
[38] Zeitouni, O., Approximate and limit results for nonlinear filters with small observation noisethe linear sensor and constant diffusion coefficient case, IEEE Trans. Automat. Control, 33, 595-599 (1988) · Zbl 0647.93067
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