×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[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
[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
[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
[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
[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
[25] Takahashi, A., An asymptotic expansion approach to pricing contingent claims, Asia-Pacific Financial Markets, 6, 115-151 (1999) · Zbl 1153.91568
[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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.